36 PD07 Abstracts - SIAM

36 PD07 Abstracts

CP1

Energy Functional of the Dbvp and the First Eigen-value of the Laplacian

Let B1 be a ball of radius r1 in Sn (Hn) with r1 < πfor Sn. Let B0 be a smaller ball of radius r0 such thatB0 ⊂ B1. Let u be a solution of the problem −Δu = 1 inΩ := B1 \B0 vanishing on the boundary. It is shown thatthe energy functional E(Ω) is minimal if and only if theballs are concentric. It is also shown that first Dirichleteigenvalue of the Laplacian on Ω is maximal if and only ifthe balls are concentric.

Anisa ChorwadwalaThe Institute of Mathematical [email protected]

CP1

Singularities, Not Existing Solutions, Mac andPDE

The non existing solution of Dirichlet problem for simplebounded domain is analyzed. If the domain is a ring forLaplace equation then the solution exist for finite inner andouter radius but it can have singularity in the solution forsmall inner radius. The solution does not exist for zero in-ner radius. To obtain the physically acceptable solution themethod of additional conditions (MAC) is applied. Thenthe bounded solution can be obtained.

Igor NeygebauerAssociate ProfessorNational University of [email protected]

CP1

A Theory of Infinite-Dimensional Evans Functionand the Associated Bundle.

The purpose of this talk is to introduce a topological frame-work to treat elliptic eigenvalue problems in a channel orbounded domain in higher dimension. The main objectsare generalizations of the Evans function and the bundleconstruction which are originally used in the stability anal-ysis of traveling wave in R.

Shunsaku NiiKyushu [email protected]

Jian DengCentral University of Finance and [email protected]

CP1

Singular Limit Problem for Some Elliptic Systems

For the sharp interface problem arising in the singular limitof some elliptic systems, we prove the existence and thenondegeneracy of solutions whose interface is a distortedcircle in a two-dimensional bounded domain without anyassumption on the symmetry of the domain.

Yoshihito OshitaDepartment of MathematicsOkayama [email protected]

CP1

The Boundary Value Problems with Strong Singu-larity of Solution.

We consider the boundary value problems with strong sin-gularity. Depending on the coefficients and right-handssides of the equation and of the boundary conditions, thesolution can be infinite at some points (the case of strongsingularity, u �∈ H1(Ω)). For these problems we offer to de-fine the solution as Rν-generalized one. Such a new conceptof solution led to distinction of two classes of the boundaryvalue problems: the problems with coordinated and non-coordinated degeneration of initial date; and also madeit possible to study the existence and uniqueness of solu-tions as well as coercivity and differential properties in theweighted Sobolev spaces.

Victor RukavishnikovRussian Academy of [email protected]

CP1

Uniqueness and Asymptotic Stability for the Ra-dial Solutions of Semilinear Elliptic Equations

Less is known of the uniqueness for the radial solutions u =u(r) of the problem Δu = f(u+) = 0 in Rn (n > 2), u(ρ) =0, u′(0) = 0, besides the cases where limr→∞u(r) = 0; andfor the cases based only on the evolution of the functions

f(t) and ddt

f(t)t

. This paper proves uniqueness for the prob-lem Da + f(u+) = 0 (r > 0), u(ρ) = 0, u′(0) = 0 basedon the assumption that f ∈ C1([0,∞)) and that ρ satisfiesa boundedness condition. Furthermore, we prove asymp-totic stability for Da + φ(r)f(u+) = 0 based only on theevolution of u′(r) and u− φ(r)f(u).

Sikiru Adigun SanniUniversity of Uyo, Uyo, [email protected]

CP2

Ultra-Fast Soliton Propagation in Nonlinear Opti-cal Media

Motivated by the recent interest in optical materialswith embedded nanostructures (so-called metamaterials),we study stability of ultra-fast pulses governed by theMaxwell-Duffing model. A solitary wave is obtained as asimulated annealing solution for the two-point boundary-value problem of the 6 by 6 system of ODEs for the trav-eling wave. Stability of solitary waves is demonstratedthrough a series of DNS as well as through a numericallinearized stability analysis.

Yevgeniy Frenkel, Victor RoytburdDepartment of Mathematical [email protected], [email protected]

CP2

Evaluation of the Finite Impulse in Plastic-ElasticWave Propagation

We consider a wave picture in semi-infinite bar in case ofplastic-elastic loading and elastic unloading. The stress inthe endpoint of the bar represents the finite impulse. Theboundary between loading and unloading zones is calledthe unloading wave. In case under consideration the initial

PD07 Abstracts 37

part of the unloading wave is the strong wave, but a jumpof strains decreases. We obtain the condition for impulseamplitude, such that this jump equals zero within finitetime. After this jump has equaled zero the unloading wavebecomes the weak wave. We obtain the form of this waveanalytically.

Alexander A. Kosmodemyanskiy

MIIT (Moscow Institute of Railways)Comp. Math [email protected]

CP2

A Variational Approach to Inverse Recovery of Co-efficient Functions in the Acoustic Wave Equation

We present a full waveform, fully nonlinear minimizationapproach for the reflection seismology problem of recover-ing the subsurface density ρ, sound speed κ, and absorptionγ from direct measurements of pressure over a part of thewater surface. Our approach consists of transforming theunderlying hyperbolic partial differential equation

1

ρ(x)κ2(x)(ptt + γ(x)pt) −∇.

1

ρ(x)∇p = F (x, t)

to the self-adjoint elliptic equation

−∇.(A(x)∇u) + (λ2B(x) + λC(x))u = −e−λ(λR(x) + S(x))

+ F (x, λ)

via the finite Laplace transform and minimizing a func-tional that is believed to possess a unique stationary point.The minimization procedure is based on a conjugate gra-dient descent algorithm using Sobolev gradients.

Ian W. KnowlesUniversity of Alabama at BirminghamDepartment of [email protected]

Mandar S. KulkarniDepartment of MathematicsUniversity of Alabama at [email protected]

CP2

Discrete Nonlinear Waves As Solutions of LinearWave Equation

A new method is described to compute thin pulse solu-tions to the linear wave equation, discretized on a uniformlattice. Unlike conventional Eulerian numerical solutions,these waves remain narrow, several grid cells (h) in width,and do not spread due to discretization error even overindefinite distances. These “pulses” represent a travelingwave front as a codimension 1 surface in the limit h-¿0. Thepulse is essentially a nonlinear solitary wave that “lives” onthe lattice. It accurately represents the total amplitude ofan arriving pulse at each lattice point in one or three di-mensions, as well as the arrival time. Multiple arrival timescan also be computed.

Subhashini [email protected]

John SteinhoffUTSI

Goethert [email protected]

CP2

Numerical Simulation on Optimal Quantum Sys-tem Control

Amazing quantum world is the most frontier realm. Amongthis, quantum system control, with its supprisingly devel-oping and rapidly growing, is extensively demanded area.It attracts numerous attention of worldwide researchersand scientists. Fortunately, laboratory simulation providesthe essential step to realize quantum control in advancedlaser technology. With this motivation, the novel point ofthis work focus on exploring substantial stage in the direc-tion of real quantum control via computational experiment.

Quan-Fang WangMechanics and Automation EngineeringThe Chinese University of Hong [email protected]

CP3

The Stability and Stabilization of Solutions of theNonlinear Differential Equations And Their Appli-cations

The presentation consists of the three parts. In the firstpart we give a review of the works of the author devotedto Liaponov stability of the motion. Given new criterion’sof stability of nonlinear differential and difference equa-tions in Banach spaces, of stability of solution of nonlinearsystems of differential equations with lateness, nonlinearsystems of differential equations with a small parameter at-tached to derivative, nonlinear systems of partial differen-tial equations. Stability of solution of nonlinear systems ofdifferential equations with discontinuous right-hand sidesis investigated also. We investigate the Turing stabilityof solution of partial differential equations. Criterion’s ofLiapunov stability are applied to regular case and to allpossible critical cases simultaneously. In the second partof the presentation we give some new criterion’s of stabi-lization of solutions of differential equations. In the thirdpart of the presentation we give applications of these crite-rions to some tasks of physics, ecology and economics. Inparticular, we offer new criterion of stability of Hotelling-Scellam models in ecology and economics, dissipative andSchrodinger systems. New criterion of stability of Kol-mogorov model in ecology are given too.

Ilya V. [email protected]

CP3

The Extremal Solutions of Nonlinear SingularBoundary Value

We consider the nonlinear boundary value problems forpartial differential equations, that are not satisfy the co-ercivity condition and for which there are not the globalsolvability results. By using the method of the insertion ofspurious control function the given problem is regularized.For each value of control functions the obtained problemhas at least one global solution. We form the discrepancyfunctional, then we prove some theorems for existence ofoptimal control. The obtained solutions are regularizators

38 PD07 Abstracts

for original problem.

Victor IvanenkoInstitute for Applied and System Analysis NAS [email protected]

Pavlo KasyanovKyiv National Taras Shevchenko University, [email protected]

Valeriy MelnikInstitute for Applied and System Analysis NAS ofUkraine, [email protected]

CP3

The Multivalued Penalty Method for EvolutionVariational Inequalities with 0-PseudomonotoneMultivalued Maps

We consider the evolution variational inequalities with 0-pseudomonotone maps. The main properties of the givenmaps have been investigated. By using the finite differ-ences method, the strong solvability for the class of evolu-tion variational inequalities with 0-pseudomonotone maphas been proved. Due to the penalty method for multi-valued maps, the existence of weak solutions for evolutionvariation inequalities on closed convex sets has been shown.The class of multivalued penalty operators has been con-structed. We have been considered one model example toillustrate the given theory.

Victor IvanenkoInstitute for Applied and System Analysis NAS [email protected]

Pavlo KasyanovKyiv National Taras Shevchenko University, [email protected]

Luisa ToscanoSecond University of Naples, Aversa, [email protected]

CP3

Krylov Subspace Spectral Methods for HyperbolicSystems

It has been demonstrated that for time-dependent variable-coefficient scalar equations, Krylov subspace spectralmethods are capable of simultaneously achieving high-order accuracy in time, along with remarkable stability,considering that they are explicit methods. In this talk,we explore the generalization of these methods to hyper-bolic systems of PDE. We also examine the use of homoge-nization via unitary similarity transformations to enhanceaccuracy and stability even further.

James V. LambersStanford UniversityDepartment of Energy Resources [email protected]

CP3

Boundary Integral Approach for a Problem in Lin-

ear Viscoelasticity

In this paper it will be studied the boundary integralformulation of the initial-boundary value problem of aVolterra type integro-differential equation

∂u(x, t)

∂t− μΔu(x, t) −

∫ t

0

(t− s)α−1

Γ(α)Δu(x, s)ds = 0,

in Ω u = f,

on ∂Ω u(x, 0) = 0.

We present the boundary integral formulation of the prob-lem by using the fundamental solution of the integro-differential operator in question. The mapping propertiesof the boundary integral operators in anisotropic Sobolevspaces will be given. Furthermore we show the validity ofthe Garding inequality. These properties are needed in theanalysis of the numerical approximation methods. Finallywe will show that the Galerkin and collocation methodsare stable and convergent.

Keijo M. RuotsalainenProfessor, University of [email protected]

CP3

Reduction of Dynamical Systems Under StochasticParametric Forcing

A novel model reduction strategy for stochastic ODE/PDEsystems with parametric forcing is discussed and illustratedwith a SEIR system. In particular: 1) We show howMarkov multiplicative parametric noise can be accuratelymodeled using additive noise provided non static effects aretaken into consideration in estimating its variance, in con-trast with the standard approach (Freidlin and Wentzell,Skorokhod). 2) We present results on how the nature of thestochastic/harmonic forcing affects the dynamics in variousparameter regimes.

Bruno D. WelfertArizona State UniversityDepartment of Mathematics & [email protected]

Reynaldo Castro, Juan LopezArizona State [email protected], [email protected]

CP4

Blade-Vortex Interaction Analysis Using PotentialFlow Theory

Abstract not available at time of publication.

Marcel IlieCarleton UniversityDepartment of Mechanical and Aerospace [email protected]

CP4

Exact Solution of the 3D Linear Eulers EquationsAs a Benchmark Test

We obtain an exact solution of the three-dimensional linearEulers equations containing a combination of entropy, vor-ticity, and acoustic waves in a mean flow. The solution is agood benchmark test for numerical schemes developed for

PD07 Abstracts 39

Computational Aeroacoustics problems. We examine theaccuracy of a sliding mesh interface method for problemsinvolving moving geometry. The sliding interface methoduses an optimized interpolation scheme with coefficientsobtained by minimizing interpolation error in wave num-ber space.

Philip P. LepoudreFlorida State [email protected]

CP4

Stable Interface Conditions for DiscontinuousGalerkin Approximations of the Navier-StokesEquations

The lecture presents a methodology for constructingwell-posed boundary and interface conditions for one-dimensional Discontinuous Galerkin approximations ofadvection-diffusion equations. While Riemann solvers havetraditionally been used at element interfaces to computethe interface flux for the case of pure advection, most dis-cretizations of the Navier-Stokes equations use an averageof the viscous fluxes on the two sides of an interface. Re-quiring the norm of the solution to not increase in time, aset of boundary/interface conditions that can be thoughtof as ”viscous” Riemann solvers can be devised, which arecompatible with the pure advection limit.

Arunasalam RahunanthanDepartment of Mathematics, University of WyomingLaramie, Wyoming [email protected]

Dan StanescuDepartment of Mathematics, University of Wyoming,Laramie, Wyoming [email protected]

CP4

The PDE2D Collocation Finite Element Method

PDE2D (www.pde2d.com, sold through Visual Numerics’e-commerce site) is a finite element program which solvesvery general PDE systems in 1D intervals, arbitrary 2D re-gions and a wide range of simple 3D regions. Both Galerkinand collocation options are available for 1D and 2D prob-lems, only collocation for 3D problems. The collocationalgorithms have some novel features, particularly relatingto how non-rectangular regions are handled. Some advan-tages and disadvantages of these collocation algorithms arediscussed, relative to the more standard Galerkin meth-ods. A new GUI now allows extremely easy access to thePDE2D collocation methods, in fact, the new GUI is im-possible to beat in terms of ease-of-use because it requiresthe user to supply the absolute minimum of informationnecessary to describe the PDEs and boundary conditions.

Granville SewellMathematics Dept.University of Texas El [email protected]

CP4

Accurate and Robust Transmissibility Upscaling onUnstructured Grids

In this talk, we consider coarse-scale modeling of flow in

porous media consisting of geological features, such asfaults and fractures, that are challenging from a grid-ding standpoint. We demonstrate an approach for ob-taining an unstructured grid, and accompanying finite vol-ume scheme, using spatially-varying multi-point flux sten-cils computed from local fine-scale solutions in such a wayas to simultaneously achieve accuracy and robustness.

Elizabeth J. SpiteriChevron Energy Technology [email protected]

James V. LambersStanford UniversityDepartment of Energy Resources [email protected]

CP5

A Positivity Preserving Central-Upwind Schemefor Chemotaxis and Haptotaxis Models

The paper is concerned with development of a new finite-volume method for a class of chemotaxis models and for aclosely related haptotaxis model. In its simplest form, thechemotaxis model is described by a system of nonlinearPDEs: a convection-diffusion equation for the cell densitycoupled with a reaction-diffusion equation for the chemoat-tractant concentration. The first step in the derivation ofthe new method is made by adding an equation for thechemoattractant concentration gradient to the original sys-tem. We then show that the convective part of the result-ing system is typically of a mixed hyperbolic-elliptic typeand therefore straightforward numerical methods for thestudied system may be unstable. The proposed methodis based on the application of the second-order central-upwind scheme, originally developed for hyperbolic sys-tems of conservation laws in [A. Kurganov, S. Noelle,and G. Petrova, SIAM J. Sci. Comput., 21 (2001),pp. 707–740], to the extended system of PDEs. We showthat the proposed central-upwind scheme is positivity pre-serving, which is a very important stability property ofthe method. The scheme is applied to a number of two-dimensional problems including the most commonly usedKeller-Segel chemotaxis model and its modern extensionsas well as to a haptotaxis system modeling tumor inva-sion into surrounding healthy tissue. Our numerical resultsdemonstrate high accuracy, stability, and robustness of theproposed scheme.

Alina ChertockNorth Carolina State UniversityDepartment of [email protected]

Alexander KurganovTulane UniversityDepartment of [email protected]

CP5

Traveling Waves in Reaction-Diffusion Equationswith Density-Dependent Diffusion

We consider a coupled system of reaction-diffusion equa-tions with density-dependent diffusion, which arises as amodel of spreading of a bacterial population in a surfaceenvironment. Traveling wave solutions of the system andtheir asymptotical behaviors are investigated. We furtherstudy the traveling wave speed dependence on various pa-

40 PD07 Abstracts

rameters. Numerical examples are presented as an illus-tration of the above results. The results are in agreementwith experimental data.

Haiyan WangArizona State [email protected]

Zhaosheng FengUniversity of Texas-Pan [email protected]

CP5

Global Stability for Lotka-Volterra Systems

We study Cauchy problem for the n-species Lotka-Volterratree systems of reaction-diffusion equations. We obtain aset of sufficient conditions for the globally asymptotic sta-bility of the solutions to Cauchy problem . The criteriain this paper are in explicit forms of the parameters, andthus, are easily verifiable. Moreover, this criteria is appli-cable to competition model, cooperation model, as well asto predator-prey model.

Xinhua JiInstitute of MathematicsAcademia [email protected]

CP5

A Competition Model for Two Invasive Species anda Native Species

The dynamics of three competing species is examined inspace-time. The model is a system of Lotka-Volterra typepartial differential equations, with one equation lacking adiffusion term. The system is analyzed in terms of travelingwaves, including conditions on the propagation speed ofthe wave front. Comments include material applicable toecological invasions by multiple species.

Daniel L. KernUniversity of Nevada, Las VegasDept of Mathematical [email protected]

CP5

Dependence of the Speed of Traveling Wave onInvader Species and Dynamics in Two-Prey, One-Predator Commensal Effect Model

We propose the relation between invader species and thediffusion speed of traveling wave in three species prey-pedator model. It is important problem biologically whataffects the spreding speed (diffusion speed) by invasion ofoutsider species. We apply the problem to two-prey, one-predator model involving the effect of carcasses. The modelhas a commensalism interaction that resident prey specieseat the remains of carcasses of invader species after preda-tor species have consumed it. Under some biological as-sumptions, we propose the model as ODE system and alsoconsider the spatial diffusion effect as PDE system. ThePDE system is given as the following:

∂∂th1 = d1

∂2

∂x2 h1 + e1

(1 − h1

k1

)h1 − α1h1p + ωh2ph1,

∂∂th2 = d2

∂2

∂x2 h2 + e2

(1 − h2

k2

)h2 − α2h2p,

∂∂tp = d3

∂2

∂x2 p− γp + β1h1p + β2h2p.

For the ODE system, we prove permanence as well as uni-form boundedness and show that the effect of the car-casses leads to chaotic dynamics for biologically reasonablechoices of parameters by numerical simulations. Then weconsider the spatial diffusion problem as the PDE systemand propose that the diffusion speed depends only on thediffusion coefficient of ’invader species’.

Sungrim S. LeeGraduate school of environmental sciencesOkayama [email protected]

CP5

Anguilliform Swimming Through Activation in AnElastic Rod

I will discuss a model for anguilliform swimming that in-corporates a simple model for muscle forces involving therelease of calcium ions from the sarcoplasmic reticulum,and coupled to an activated elastic rod. Most fish swim byrhythmically passing a wave of muscle activation from headto tail, which creates a corresponding wave of curvature. Iwill provide an explanation for the fact that the wave ofactivation travels faster than the wave of curvature.

Tyler McMillenCalifornia State University, FullertonDepartment of [email protected]

Philip HolmesPrinceton UniversityMAE [email protected]

Thelma WilliamsSt. Georges Hospital Medical [email protected]

CP6

Nondegenerate Isentropic and AnisentropicOne-Dimensional Wave-Wave Regular InteractionStructures: Details of a Parallel

We use two concurrent approaches [“algebraic” (Burnat;essential for isentropic multidimensional extensions) and“differential” (Martin)] to characterize nondegenerate one-dimensional wave-wave regular interaction structures. Theanisentropic “differential” characterization results from asignificant [Monge−Ampere] extension and is proven to bestrictly “non-algebraic” (the isentropic “differential” and“algebraic” characterizations are proven to coincide). Aparallel between two contexts [isentropic, anisentropic], fo-cused on significant contrasts between the “differential”types of wave-wave regular interaction structures respec-tively associated, is finally considered.

Marina Ileana DinuPolytechnical University of Bucharest, Dept. of Math. [email protected]

Liviu Florin DinuInstitute of Mathematics of the Romanian [email protected], [email protected]

PD07 Abstracts 41

CP6

Coupled Mode Theory of Resonant Scattering ofWater Waves by a Two Dimensional Array of Ver-tical Cylinders

We study the Bragg resonance of surface water waves by atwo-dimensional array of vertical cylinders covering a largearea of the sea. Using the resonance criterion known inthe solid- state and crystallography, we employ asymptotictechniques to derive two-dimensional coupled-mode equa-tions for the envelopes of scattered waves resonated by aplane incident wave. Explicit analytical solutions are ob-tained for a long strip of cylinder array. Examples of bothtwo-wave and three-wave resonances are discussed in de-tail. Roles of the bandgaps are examined. The analysisand results may also apply to sound or optical waves inperiodic media.

Chiang C. MeiDept of Civil and Environmental [email protected]

Yile LiShell International Exploration & ProductionHoutson [email protected]

CP6

Wave-Breaking Dissipation in Wave/Current In-teractions

Wave breaking dissipates energy in oceanic waves and it isagreed that it affects currents as well. The question to beanswered are what aspects of the dynamics of waves, cur-rents and their interactions are affected and how. Usingmulti-scale ideas and a stochastic parametrization of theLagrangian fluid path it is possible to model the determin-istic interactions and include in these dissipative effects.The resulting model is then capable of answering some ofthe phenomenological questions, and further, it can be usedto make specific and testable predictions. The talk will fo-cus on the modeling strategy and on preliminary resultsderived from the analysis of the model.

Juan M. RestrepoUniversity of [email protected]

CP6

Standing Wavesfor a Generalized Davey-Stewartson System: Re-visited

The existence of standing waves for a generalized Davey–Stewartson (GDS) system was shown in [A. Eden and S.Erbay, Standing waves for a generalized Davey–Stewartsonsystem, J. Phys. A: Math. Gen. 39 13435–13444 (2006)]using an unconstrainted minimization problem. Here, weconsider the same problem but relax the condition on theparameters to χ+ b < 0 or χ+ b

m1< 0. Our approach is to

use a constrained minimization problem and utilize Lions’concentration compactness lemma. When both methodsapply we show that they give the same minimizer and weobtain a sharp bound for a Gagliardo–Nirenberg type in-equality. This leads to a global existence result for small-mass solutions. Moreover, we show that when p > 2, theLp-norms of solutions to the Cauchy problem for a GDS

system converge to zero as t → ∞.

Ihsan A. TopalogluIndiana [email protected]

Alp EdenBogazici UniversityFeza Gursey [email protected]

CP7

Numerical Analysis for the Nonlocal Allen-CahnEquation

We propose a stable, convergent and linear finite differ-ence scheme to solve numerically the nonlocal Allen-Cahnequation which can model a variety of physical and biolog-ical phenomena involving media with properties varying inspace. Also we prove that the scheme is uniquely solvableand the numerical solution will approach the true solutionin the L∞ norm.

Sarah M. Brown, Jianlong HanSouthern Utah Universitybrown [email protected], [email protected]

CP7

Title Not Available at Time of Publication


Ming ChenUniversity of [email protected]

CP7

Multiscale Phase Field Modeling of MartensiticMicrostructure Evolution and Nucleation

A three-dimensional Landau theory for multivariant stress-induced martensitic phase transformations was recently de-veloped (V. I. Levitas, D. L. Preston and D. W. Lee, Phys.Rev. B, 134201, 2003) This new theory describes correctlythe stress and temperature effect on PTs. The time de-pendent Ginzburg-Landau equation is used to simulate mi-crostructures evolution. The barrierless martensitic nucle-ation at various dislocation configurations (single disloca-tion, dislocation dipole and low angle grain boundary) isalso considered.

Dong-Wook LeeMechanical EngineeringTexas Tech [email protected]

CP7

The Numerical Computation of the Critical Bound-ary Displacement for Radial Cavitation

We study radial solutions of the equations of isotropic elas-ticity in two dimensions (for a disc) and three dimensions(for a sphere). This is equivalent to a nonlinear (quasi-linear) boundary value problem depending parametricallyon the boundary displacement. We describe a numericalscheme for computing the critical boundary displacementfor cavitation. We give examples for specific materials andcompare our numerical computations with some previous

42 PD07 Abstracts

analytical results. We give a characterization in the phaseplane of the nonconstant solutions of the correspondingEuler–Lagrange equations for radial solutions in terms ofthe curves of constant normal component of Cauchy stressfor general classes of stored energy functions.

Pablo NegronUniversity of Puerto Rico - [email protected]

Jeyabal SivaloganathanDepartment of Mathematical SciencesUniversity of Bath, [email protected]

CP7

Analysis of Microbial Depolymerization of Xenobi-otic Polymers Based on Mathematical Models andExperimental Data

A microbial depolymerization process falls into one of twocategories: exogenous type and endogenous type. Assum-ing that a degradation rate is a product of a molecularfactor and a time factor, a time dependent depolymeriza-tion model is transformed into a time independent model.Given the weight distribution before and after degrada-tion, the molecular factor is determined with techniquesdeveloped in the previous studies. These techniques andtechniques to determine the time factor will be discussed.

Masaji WatanabeGraduate School of Environmental Science, [email protected]

Fusako KawaiResearch Institute for BioresourcesOkayama [email protected]

CP8

Mathematical Model of Filtration in Membrane

We study a general mathematical model of permeate flux intwo variables. First, we state the fundamental differentialequation for the inner flow and the permeate flux, which areobtained from the continuity equation. The flux satisfies afirst order linear partial differential equation and the innerflow it obtained from a second order non-linear differentialequation.We solve all the equations.For the linear one, themethod of characteristics is used.

Ezio [email protected]

Carlos TarazagaUniversidad Nacional de SanLuis,San Luis,5700,[email protected]

CP8

Finite Element Method for the Plane StationaryFree Boundary Flow

We consider the stationary flow in a bounded domainΩ ⊂ R2 with boundary Γ = Γ1 ∪ Γ2. Γ1 is a fixed partof the boundary, Γ2 is an unknown free boundary. The

flow is governed by the Navier-Stokes equations. The so-lution is found by the successive approximations. Everyapproximation is the solution to the linearized problem inthe domain with known boundary. This solution is con-structed by the finite element method.

Ilya S. MogilevskiyTver universityMathematical [email protected]

CP8

An H2-Compatible Finite Element Solver for theNavier-Stokes Equations

Liu, Liu, and Pego have developed an unconditionally sta-ble algorithm for finite element discretization of the Navier-Stokes equations in the setting of the H2 Sobolev spaces.The algorithm enforces conservation of mass without aneed for the inf-sup condition that is common in H1 for-mulations. In domains with reentrant corners the solutionof the Navier-Stokes equations need not be in H2, there-fore the algorithm needs to be modified. We have modifiedthe algorithm by recasting it in weighted Sobolev spaceswhere appropriate weights at reentrant corners compensatefor the solution’s singularities. The numerical results ob-tained this way show good agreement with those obtainedby the usual H1 methods.

Ana Maria SoaneUniversity Of Maryland, Baltimore CountyDepartment of Mathematics and [email protected]

CP8

On a 1D Model of the 3D Euler/Navier-StokesEquations

In 1985 a simple 1D equation ωt = H(ω)ω is proposed by P.Constantin, P. Lax and A. Majda as a model of the vortexstreteching phenomenon in 3D incompressible Euler equa-tions. This model equation blows up in finite time. Later S.Schochet discovered a counter-intuitive phenomenon: the”viscous” model ωt = H(ω)ω + ωxx blows up faster forcertain initial values. In this talk I will present new under-standing of the interaction between stretching and dissipa-tion. With this understanding, the phenomenon discoveredby Schochet is no longer counter-intuitive.

Xinwei [email protected]

CP9

Multiscale Particle Method for Hyperbolic PDEs

A meshless particle method in a multiscale framework isinvestigated. The aim is to obtain numerical solutions forPDEs by avoiding the mesh generation and by employing aset of particles arbitrarily placed in problem domain. Theelimination of a mesh combined withthe properties of di-lation and translation of scaling and wavelets functions isparticularly suitable for problems with large deformationsand high gradients. Numerical examples are presented toillustrate the flexibility of the proposed approach.

Elisa FrancomanoDipartimento di Ingegneria InformaticaUniversita degli Studi di Palermo - Italia

PD07 Abstracts 43

[email protected]

Adele TortoriciUniversita degli Studi di Palermo - [email protected]

Elena ToscanoUniversita degli Studi di [email protected]

CP9

Non-Oscillatory Central Schemes for a Traffic FlowModel with Arrhenius Look-Ahead Dynamics

We develop non-oscillatory central schemes for a trafficflow model with Arrhenius look-ahead dynamics. Thismodel, takes into account interactions of every vehicle withother vehicles ahead (“look-ahead” rule) and can be writ-ten as a one-dimensional scalar conservation law with aglobal flux. The proposed schemes are extensions of thefirst-order staggered Lax-Friedrichs scheme and the second-order Nessyahu-Tadmor scheme, which belong to a class ofGodunov-type projection-evolution methods, but does notrequire any (approximate) Riemann problem solver, whichis unavailable for conservation laws with global fluxes. Ournumerical experiments demonstrate high resolution, stabil-ity, and robustness of the proposed method, which is usedto numerically investigate both dispersive and smoothingeffects of the global flux.

Alexander KurganovTulane UniversityDepartment of [email protected]

MS1

Vortex Solutions of a ”Semi-Stiff” Boundary ValueProblem for the Ginzburg-Landau Equation

We study solutions of 2D Ginzburg-Landau equation−Δu + 1

ε2u(|u|2 − 1) = 0 subject to the so-called ”semi-

stiff” boundary conditions: the Dirichlet condition for themodulus, |u| = 1, and the homogeneous Neumann condi-tion for the phase. The principal result of this work showsthat there are stable solutions of this problem with vorticesthat are located near the boundary and have bounded en-ergy as ε → 0. The proof is based on the introduction of anotion of the approximate bulk degree which is stable withrespect to the weak H1 convergence unlike the standarddegree boundary conditions. This is a joint work with V.Rybalko.

Leonid BerlyandPenn State UniversityEberly College of [email protected]

Volodymir RybalkoInstitute for Low Temperature PhysicsKharkov, [email protected]

MS1

Global Minimizers for a P-Ginzburg-Landau En-ergy Functional

For a p-Ginzburg-Landau energy on the plane we prove theexistence of global minimizers in the class of functions with

degree equal one at infinity.


Dmitry GolovatyThe University of AkronDepartment of Theoretical and Applied [email protected]

Yaniv AlmogLouisana State [email protected]

Itai ShafrirTechnion - [email protected]

MS1

On a Minimization Problem with a Mass Con-straint Involving a Potential Vanishing on TwoCurves

We study a singular perturbation type minimization prob-lem with a mass constraint on a domain involving a poten-tial vanishing on two curves in the plane. We analyse thelimiting behavior of both the minimizers and their energieswhen the small parameter epsilon goes to zero and demon-strate the effect of the domain geometry on this behavior.

Itai ShafrirDepartment of Mathematics, Technion- [email protected]

Nelly AndreDepartment of MathematicsUniversity of Tours, [email protected]

MS1

Superconducting Wires Subjected to Electric Cur-rents

We consider a one-dimensional time-dependent Ginzburg-Landau model for the current in a superconducting wiredriven by an applied current. We discuss the co-existenceof normal and supercurrent in the wire and the bifurcationfrom the normal state of both steady-state supercurrentsand periodic solutions possessing vortices.

Kevin ZumbrunIndiana [email protected]

Peter SternbergIndiana UniversityDepartment of [email protected]

Jacob Koby RubinsteinIndiana University and The Technion, IsraelDepartment of [email protected]

44 PD07 Abstracts

MS2

Modeling the Inhomogeneous Response in Steadyand Transient Flows of Wormlike Micellar Solu-tions

Surfactants are amphiphilic molecules that self-assemblein solution. Under certain conditions they self-assembleinto long worm-like micelles. Wormlike micellar solutionshave important uses in the petroleum industry and thehealth-care products industry. In solution these long flexi-ble wormlike structures entangle thus exhibiting viscoelas-tic properties like polymers; but, in contrast to polymers,they also break and reform continuously thus presenting anadditional relaxation mechanism. Experimental observa-tions show that these solutions exhibit distinctive (inhomo-geneous) behaviors under different deformation conditions;for example, in steady shear flow these surfactant solutionsdevelop shear bands (or more accurately shear rate bands).Concurrent with this banding is the development of a shearstress plateau in the flow curve over a broad range of shearrates. In this talk, several constitutive models (VCM, PEC,PEC+M) are investigated numerically and analytically ina cylindrical Taylor Couette geometry. These models aredescribed through coupled systems of nonlinear partial dif-ferential equaitons. The VCM model is a two- species net-work model that incorporates a discrete version of Cates’breaking and reforming dynamics. The PEC and PEC+Mmodels are, respectively, a one mode and a two mode ap-proximation of the full VCM model (without the effects ofstress-concentration coupling). The VCM model has beenstudied in viscometric (homogeneous) flow in Vasquez et.al. JNNFM.07. The rheological predictions of these con-stitutive model are investigated analytically and computa-tionally in steady and transient shear and in ”step strain”and the results are compared with experiment. The flowsdescribed are inhomogeneous and do exhibit shear band-ing. The effects of curvature, of diffusive terms in the stressequations, and of inertia are discussed. This work was sup-ported by the NSF-DMS #0405931

L. Pamela CookDepartment of Mathematical SciencesUniversity of [email protected]

Lin ZhouENSUREDMAIL, INC.Wilmington, [email protected]

Paula VasquezDept of Mathematical SciencesUniversity of [email protected]

Gareth McKinleyDept of Mechanical EngineeringMassachusetts Institute of [email protected]

MS2

Numerical Simulations of Drop Deformation forViscoelastic Liquid-Liquid Systems

We investigate the motion of the interface separating adrop and its surrounding liquid (matrix) in a large channeldriven by wall-induced shear. The governing equations arethe momentum equation, incompressibility, and constitu-tive equations for the Giesekus model. The Giesekus liquid

is a viscoelastic liquid with first and second normal stressesand shear-thinning. We implement a volume-of-fluid algo-rithm with a parabolic re-construction of the interface forthe calculation of the surface tension force (VOF-PROST).Numerical results are checked against small deformationtheory for a viscoelastic matrix liquid, an Oldroyd-B ex-tensional flow simulation, and experimental data on vis-coelastic drop/liquid systems from S. Guido and P. Molde-naers (private communication). Reasons for the differencebetween Newtonian and viscoelastic drop shapes are ex-plored with numerical simulations.

Yuriko RenardyVirginia TechDepartment of [email protected]

MS2

Singularities and Transport in Viscoelastic Flows

In the past several years it has come to be appreciated thatin low Reynolds number flow the nonlinearities providedby non-Newtonian stresses of a complex fluid can providea richness of dynamical behaviors more commonly asso-ciated with high Reynolds number Newtonian flow. Forexample, experiments by V. Steinberg and collaboratorshave shown that dilute polymer suspensions being shearedin simple flow geometries can exhibit highly time depen-dent dynamics and show efficient mixing. The correspond-ing experiments using Newtonian fluids do not, and in-deed cannot, show such nontrivial dynamics. To betterunderstand these phenomena we study numerically the 2DOldroyd-B Viscoelastic model at low Reynolds number. Abackground force is used to create a periodic cell with four-roll mill vertical structure around a hyperbolic fixed point.We consider both steady and time-periodic forcing. For lowWeissenberg number (Wi) the elastic stresses are boundedand .slaved. to the forcing, with mixing confined to smallsets near the hyperbolic point. At larger Wi an analog tothe coil-stretch transition occurs yielding large stresses andstress gradients concentrated on sets of small measure, per-haps indicating the development of singularities. The flowthen becomes very sensitive to perturbations in the forcingand there is a transition to global mixing in the fluid.

Becca ThomasesCourant InstituteNew York [email protected]

MS2

Mathematical Modeling and Computational Issues

The properties of viscoelastic fluids are governed by theflow-induced evolution of molecular configurations. In ad-dition to the usual balance equations this requires consti-tutive equations for the configuration and stress tensor.Modeling and numerical simulation of viscoelastic fluids isnot an easy task and improvements are needed in bothareas. Recently, various new constitutive equations havebeen derived from microscopic theory. We discuss sometraditional and recently proposed constitutive equations.The focus will be on mathematical modeling and compu-tational issues.

Peter WapperomDepartment of Mathematics,Virginia [email protected]

PD07 Abstracts 45

MS3

Semi-Strong Pulse Dynamics

Pulse interactions play a central role in a variety of pat-tern formation phenomena. They can be distinguished intothree types: weak interactions, in which the pulses are as-sumed to be sufficiently far apart, the fully strong interac-tions, and the intermediate concept of semi-strong interac-tions, that has been introduced in the context of singularlyperturbed systems, in which only some components of thepulse interact weakly. In this talk, an overview will be givenof the dynamics of semi-strong pulse interactions, as wellas the methods by which they can be studied analytically.

Tasso J. KaperBoston UniversityDepartment of [email protected]

Arjen DoelmanCWI Amsterdam, the [email protected]

Keith PromislowMichigan State [email protected]

MS3

Asymptotic Behavior Near Planar TransitionFronts for the Cahn-Hilliard Equation

I will discuss recent results on the stability of stationarysolutions for the Cahn-Hilliard equation in Rd, d ≥ 1.For the case d = 1, there are precisely three types ofnon-constant bounded stationary solutions, periodic solu-tions, pulse-type (reversal) solutions, and monotonic tran-sition fronts. These solutions can be categorized as follows:the periodic and reversal solutions are both spectrally un-stable, while the transition fronts are nonlinearly (phase-asymptotically) stable. The cases d ≥ 2 are more compli-cated, and I will discuss what is known about stationarysolutions in these cases. Particular emphasis will be placedon planar transition front (or “kink”) solutions.

Peter HowardTexas A&[email protected]

MS3

Pulse Dynamics in a Three-Component Model

We analyze the 3-component RD system developed bySchenk, Or-Guil, Bode, and Purwins. The system consistsof bistable activator-inhibitor equations with an additionalinhibitor that diffuses more rapidly than the standard in-hibitor. It is a prototype 3-component system that gen-erates rich pulse dynamics and interactions. We establishthe existence and stability of stationary 1-pulse and 2-pulsesolutions, as well as of travelling 1-pulse solutions. Also,we analyze various bifurcations, including the SN bifurca-tions in which they are created, as well as the bifurcationfrom a stationary to a travelling pulse. For 2-pulse solu-tions, we show that the third component is essential, sincethe reduced bistable 2-component system does not supportthem. Finally, as we illustrate with numerical simulations,these solutions form the backbone of the rich pulse dynam-ics this system exhibits, including pulse replication, pulseannihilation, breathing pulses, and pulse scattering, among

others.

Tasso J. KaperBoston UniversityDepartment of [email protected]

Arjen DoelmanCWI Amsterdam, the [email protected]

Peter van [email protected]

MS3

A Numerical Method for Coupled Surface andGrain Boundary Motion

We study the coupled surface and grain boundary motionin a bi-crystal in the context of the “quarter loop” geom-etry. Two types of normal curve velocities are involved inthis model: motion by mean curvature and motion by sur-face diffusion. Three curves meet at a single point wherejunction conditions are given. A numerical formulationthat describes the coupled normal motion of the curves andpreserves arc length parametrization up to scaling is pro-posed. The formulation is shown to be well-posed in a sim-ple, linear setting. Equations and junction conditions areapproximated by finite difference methods. Numerical con-vergence to exact traveling wave solutions is shown. Otherformulations of the problem and their numerical propertiesare discussed.

Brian R. Wetton, Zhenguo PanUniversity of British ColumbiaDepartment of [email protected], [email protected]

MS4

Compactness for the Landau State in Thin FilmsMicromagnetics

We study a model for the magnetization in thin ferromag-netic films. It comes as a variational problem, depend-ing on two parameters η and ε, for S2-valued maps m(the magnetization). We are interested in the behaviorof minimizers as η, ε → 0. The minimizers are expectedto concentrate on lines at a scale η with an energy of or-der O(1/η| log η|) and around a vortex at a scale ε with anenergy of order O(| log ε|). We prove compactness of mag-netizations in the regime where a vortex energy is moreimportant than a line energy.

Radu Ignat

Universite Paris 11 (Orsay)[email protected]

Felix OttoUniversity of [email protected]

MS4

2-D Stability of the Neel Wall

We are interested in thin–film samples in micromagnetism,where the magnetization m is a 2–d unit–length vector

46 PD07 Abstracts

field. More precisely we are interested in transition layerswhich connect two opposite magnetizations, so called Neelwalls. We prove stability of the 1–d transition layer under

2–d perturbations. This amounts to the investigation ofthe following singularly perturbed energy functional:

E2d(m) = ε

∫|∇m|2 dx +

1

2

∫|∇−1/2∇ ·m|2 dx.

The topological structure of this two–dimensional problem

allows us to use a duality argument to infer the optimallower bound. The lower bound relies on an ε–perturbationof the following logarithmically failing interpolation in-equality ∫

|∇1/2φ|2 dx �≤ C sup |φ|∫

|∇φ| dx.

Hans KnuepferUniversity of [email protected]

MS4

Z-Invariant Micromagnetic Configurations inCylindrical Domains

In this presentation, we have proposed a rather completestudy of the micromagnetic model when the configurationsare assumed to be vertically invariant, the (fixed) domainis cylindrical and when the exchange constant vanishes to0. The result is given as a Γ− convergence theorem andwhen the external magnetic field has a small enough mag-nitude, it fully explains the Van den Berg construction.Moreover, the hypothesis of a small exterior field enablesus to catch the (form of) the next order term in energy.This term is achieved with a kind of Bloch walls instead ofNeel walls for thin films. From a physical point of view, itis known that typical walls in thick films are the so-called“assymetric Bloch walls” which consist in a Bloch wall in-side the thickness of the sample closed by Neel caps onthe top and bottom parts of wall. Thus, our results mayseem physically meaningless. On the other hand, the nu-merical computations given in show that in thick films, themagnetization is vertically invariant in the core part of thewall and the top and bottom Neel caps are small in thick-ness. Hence, our model could well explain the core partof the (asymetric) Bloch wall, the Neel parts being caughtenergetically by the fact that our configurations are nottangent to the (horizontal) boundaries on the top and bot-tom parts. Indeed, if most of the configuration is verticallyinvariant, our ansatzes become relevant.

Stephane LabbeUniversite Paris-Sud [email protected],fr

MS4

Slow Motion of Gradient Flows

Sometimes physical systems exhibit “dynamic metastabil-ity,” in the sense that states get drawn toward so-calledmetastable states and are trapped near them for a longtime. A familiar example is the one-dimensional AllenCahn equation. In this work, we give sufficient conditionsfor a gradient flow system to exhibit metastability. We

then apply the abstract result to give a new analysis of

the 1-d Allen Cahn equation. The central ingredient is anonlinear energy-energy-dissipation relationship.

Maria G. WestdickenbergGeorgia Institute of [email protected],edu

Felix OttoUniversity of [email protected]

MS5

Numerical Stencils for Large Scale Wave Propaga-tion Problems

We have developed numerical schemes for the acoustic waveequation that minimize the dispersion error. To controlcomplexity in large 3D computations we are forced to useonly a few grid points per wavelength. The resulting dis-persion is a larger source of error than order. In geophysics,the wave equations have discontinuous velocity and den-sity functions, for instance, at the water-solid interface andalso rock-salt interface. When the numerical stencil crossesa discontinuity, we need to consider modifications to theweights in the stencil, so that we get the correct reflectioncoefficients at the discontinuities. We will review analyticand numerical results.

Phillip BordingMemorial Univeristy of NewfoundlandDepartment of Earth [email protected]

Andreas AtleDept of Earth SciencesMemorial University of [email protected]

MS5

Maximization of the Quality Factor of an OpticalResonator

We consider resonance phenomena for the scalar waveequation in an inhomogeneous medium. Resonance is asolution to the wave equation which is spatially localizedwhile its time dependence is harmonic except for decaydue to radiation. The decay rate, which is inversely pro-portional to the qualify factor Q, depends on the materialproperties of the medium. In this work, the problem of de-signing a resonator which has high Q (low loss) is consid-ered. High Q resonators are desirable in a variety of appli-cations, including photonic band gap devices. We demon-strate a numerical optimization approach to find high Qstructures.

Chiu-Yen KaoDept. of MathematicsOhio State [email protected]

Fadil SantosaSchool of MathematicsUniversity of [email protected]

MS5

A Fast Phase Space Method for Computing Creep-

PD07 Abstracts 47

ing Waves

Creeping waves can give an important contribution to thesolution of medium to high frequency scattering problems.They are generated at the shadow lines of the illuminatedscatterer by grazing incident waves and propagate alonggeodesics on the scatterer surface, continuously sheddingdiffracted waves in their tangential direction. In this talkwe show how the wave propagation problem can be formu-lated as a partial differential equation (PDE) in a three-dimensional phase space. To solve the PDE we use a fastmarching method. The PDE solution contains informa-tion about all possible creeping waves. This informationincludes the phase and amplitude of the field, which areextracted by a fast post-processing. Computationally thecost of solving the PDE is less than tracing all rays individ-ually by solving a system of ordinary differential equations.We consider an application to mono-static radar cross sec-tion problems where creeping waves from all illuminationangles must be computed. The numerical results of the fastphase space method and a comparison with the results ofray tracing are presented.

Olof RunborgKTH, Stockholm, [email protected]

Mohammad MotamedDepartment of Computer Science and CommunicationRoyal Institute of Technology (KTH)[email protected]

MS5

Efficient Numerical Methods Based on WienerChaos Expansion for Stochastic Maxwell Equations

We propose a new stochastic model for general spatiallyincoherent sources with applications in photonic crystal.The model naturally incorporates the incoherent propertyand leads to stochastic Maxwell equations. Fast numeri-cal methods based on Wiener Chaos Expansions (WCE),which convert the random equations into coupled system ofdeterministic equations, are developed. The new methodscan achieve 2 order of magnitude faster computation timeover the standard method in photonic crystal spectrometerdesign.

Haomin ZhouGeorgia TechSchool of [email protected]

MS6

Fluid-Structure Interaction in Blood Flow: AnOverview

The focus of this minisymposium and of this talk will beon the analysis and computation of fluid-structure inter-action in blood flow. Understanding solutions to moving-boundary problems describing fluid-structure interactionbetween blood flow and arterial walls is important in under-standing the mechanisms leading to various complicationsin cardiovascular function. Although fascinating progresshas been made in some areas of modeling and simulationof the human cardiovascular system many of the basic dif-ficulties remain open and will continue to present majorchallenges in the years to come. The speaker will give anoverview of the main problems and difficulties associatedwith the study of fluid-structure interaction in blood flow.

Recent results in the analysis of solutions to the benchmarkproblem in blood flow will be presented and recent develop-ments in the numerical algorithm design will be mentioned.Applications involving certain cardiovascular interventionswill be shown. Colalborators: Dr. Z. Krajcer and Dr.D. Rosenstrauch (Texas Heart Institute), Dr. C. Hartley(Baylor College of Medicine), Prof. R. Glowinski, Prof.T.W. Pan, Prof. G. Guidoboni (University of Houston),Prof. A. Mikelic (University of Lyon 1, FR), Prof. J. Tam-baca (University of Zagreb, CRO)

Suncica CanicDepartment of MathematicsUniversity of [email protected]

MS6

Proof of the Existence of a Solution to a Fluid-Structure Interaction Problem in Blood Flow

We prove the existence of a solution to a free bound-ary problem modeling blood flow in viscoelastic arter-ies.The model equations are obtained using asymptoticreduction from the incompressible, viscous Navier-Stokesequations,coupled with the linearly viscoelastic membraneequations. The reduced, effective equations are defined onthe cylindrical geometry. The resulting problem is nonlin-ear with a free boundary. The existence of a weak solutionis obtained by using the fixed point arguments.

Taebeom KimDepartment of MathematicsUniversity of [email protected]

MS6



Alfio QuarteroniEcole Pol. Fed. de [email protected]

MS6

A Fictitious Domain Method for Simulating Vis-cous Flow in a Constricted Elastic Channel Subjectto a Uniform External Pressure

Cardiovascular illness is most commonly caused by a con-striction, called a stenosis. A nonlinear mathematicalmodel with a free moving boundary is used to model elasticwall with constriction subjected to a uniform external pres-sure and a prescribed pressure drop. A fictitious domainmethod combined with operator splitting techniques andfinite element method has been developed to simulate vis-cous incompressible flow in the above channel. Numericalresults will be presented in this talk.

Tong WangDepartment of MathematicsUniversity of [email protected]

MS7

Free Surface Problems in Irrotational 2D and 3D

48 PD07 Abstracts

Fluids

We consider the irrotational water wave and the vortexsheet. We describe a formulation of the problems usingspecial parameterizations and convenient variables. In 2D,the parameterization is by arclength and in 3D the pa-rameterization is isothermal. The formulation leads to aquasilinear hyperbolic system of evolution equations for thevortex sheet (with surface tension) or for the water wave(either with or without surface tension). Having writtenthe evolution equations in this way, we are then able tomake energy estimates. In the water wave case, these es-timates are uniform in surface tension, allowing us to takethe limit as surface tension tends to zero. This gives a newproof of well-posedness of the irrotational water wave in3D. Most of this is joint work with Nader Masmoudi. Ifpossible, there will be some discussion of singularity for-mation for some of these problems.

David AmbroseClemson [email protected]

MS7

Well-Posedness of the Free-Boundary Compress-ible Euler Equations in Vacuum

We prove well-posedness for the free-boundary compress-ible Euler equations in vacuum. In this setting, the densityvanishes on the boundary, and leads to a degenerate non-linear wave equation for the Jacobian determinant of theLagrangian coordinate. This is joint work with D. Coutandand H. Lindblad.

Steve ShkollerUniviversity of California, DavisDepartment of [email protected]

MS7

Effect of Vorticity on Steady Water Waves

We compute 2D finite-depth steady periodic water waveswith general vorticity and large amplitude. In particular,the effect of a shear surface layer, which could be producedby the action of wind, is investigated. We find that themaximum amplitude increases as a function of vorticity.This occurs along the connected set of water waves up to apossible stagnation point in the interior or at the bottom.

Walter Strauss, Joy KoBrown [email protected], [email protected]

MS7

Free Boundary Problems of the Euler Equation andHydrodynamical Instabilities

We consider the evolution of free surfaces of incompress-ible and invicid fluids. Neglecting the gravity, we are in-terested in the cases of 1.) the motion of a droplet in thevacuum with or without surface tension and 2.) the mo-tion of the interface between two fluids with surface ten-sion. The evolution of these fluid boundaries and the ve-locity fields is determined by the free boundary problemof the Euler’s equation. Each of these problems can beconsidered in a Lagrangian formulation on an infinite di-mensional Riemannian manifold of volume preserving dif-feomorphisms. In the absence of surface tension, the well-

known Rayleigh-Taylor and Kelvin-Helmholtz instabilitiesappear naturally related to the signs of the curvatures ofthose infinite dimensional manifolds. The surface tensionproduces stronger conservative forces than the instabili-ties and thus regularizes the surface evolution. A scale offunctionals as ”energies” are defined and they bound highSobolev norms of the velocity field as well as the meancurvature of the fluid boundary. Thus we establish theregularity of the solutions for a short time depending onthe initial data. In the one fluid case, the energy esti-mates is uniform as the surface tension tends to zero ifthe Rayleigh-Taylor sign condition is satisfied. This justi-fies the zero surface tension problem as the limit of smallsurface tension problem.

Chongchun ZengGeorgia [email protected]

Jalal ShatahNew York [email protected]

MS8

AeroElastic Flutter Boundary Expansion withBoundary Control

No aeroelastic system is stable; and cannot be be stabilizedwith controls (Linear or Non-Linear) on the structure. Thepossibility of increasing the flutter speed with self-strainingactuators is discussed.

A. V. BalakrishnanUniversity of California, Los AngelesFlight System Research [email protected]

MS8

Boundary Control and Stability in Dynamic FluidStructure Interaction Models

Fluid structure interaction comprising of Navier-Stokesand dynamic system of elasticity is considered . The cou-pling holds on interface between fluid and solid and is pre-scribed via matching conditions imposed on Cauchy Polyastress tensors and velocities. Problems to be discussed are:Stability and boundary uniform stabilization of the non-linear model. Optimal Bolza boundary control problemdefined for linearization of the model Optimal rates of sin-gularity (blow up) of controls and of transfer function willbe given.

Irena LasieckaUniversity of Virginia, CharlottesvilleDepartment of [email protected]

MS8

Nonlocal Models of Transport in Multiscale PorousMedia

We discuss ”usual” model for advection-diffusion-dispersion in porous media and its validity in the presenceof multiple scales especially when they are not well sepa-rated. Problem is motivated by recent experimental resultsby Zinn, Haggerty et al. We present a new model based onvariational duality principles extending the classical dou-ble porosity models. Model preserves mass and has discrete

PD07 Abstracts 49

and continuous variants. Some computational realizationsemphasizing transition between scales and nonlocal effectssimilar to non-Fickian dispersion will be given.

Malgorzata Peszynska, Ralph ShowalterDepartment of MathematicsOregon State [email protected],[email protected]

MS8

Possio Integral Equation in Aeroelasticity (Asymp-totical Version)

The Possio integral equation in Aeroelasticity relates thepressure distribution over a slender wing in subsonic com-pressible air flow to the normal velocity (downwash). De-rived by C.Possio in 1938, it is crucially important in sta-bility analysis. In spite of enormous amount of numericalpapers with strong results, the fundamental problem ofsolvability of the equation has not been solved. We reportpreliminary results towards solvability.

Marianna ShubovUniversity of New HampshireDepartment of Mathematics and [email protected]

MS9

Quantum Fokker-Planck Models: Global Solutions,Steady States, and Large-Time Behavior

Dissipative open quantum systems like the quantum-Fokker-Planck (QFP) model are important for quantumBrownian motion and numerical simulations of nano-semiconductor devices. The evolution equation can bewritten in the Wigner phase-space framework or equaiva-lently for density matrix operators. We discuss global-in-time wellposedness of the nonlinear QFP-Poisson modelusing density matrices, steady states of the linear QFPequation in a given confinement potential in the Wignerframework, and establish large-time convergence (with anexponential rate).

Anton ArnoldVienna University of Technology (Austria)[email protected]

MS9



Maria GualdaniThe University of Texas at [email protected]

MS9

Semiclassical Transport Models in Thin Slabs

Quantum transport in simulation domains with a smallaspect ratio is often approximated by semiclassical limitequations, so called sub - band models, which assume thatthe transport picture is classical only in one direction. Thistalk is concerned with sub - band models in the presence ofstrong forces in the quantum mechanical direction. Aftergiving an overview of the general approach to sub - bandmodeling, we derive and discuss a system of coupled sub

- band drift diffusion equations with an inter - band col-lision operator which dissipates the quantum mechanicalequivalent of a thermodynamic entropy.

Christian RinghoferArizona State [email protected]

MS10

Electrical Impedance Tomography with ResistorNetworks

We present an inversion method that handles the se-vere ill-posedness of Electric Impedance Tomography byproper sparse parametrization of the unknown conductiv-ity. Specifically, we consider finite volume grids, whichare resistor networks in disguise, where the resistors areaverages of the conductivity over grid cells. By interpret-ing the resistor network that matches the measurements assuch discretization we are able to efficiently estimate theconductivity. Our method can also incorporate a prioriinformation about the solution, if available.

Liliana BorceaRice UniversityDept Computational & Appl [email protected]

Vladimir DruskinSchlumberger-Doll [email protected]

Fernando Guevara VasquezStanford [email protected]

MS10

Solving the Electrical Impedance TomographyProblem with Optimization Techniques

Electrical Impedance Tomography is an inverse problemof determining the conductivity function inside a conduc-tive domain from the simultaneous measurements of volt-ages and currents on the boundary of the domain. Theproblem is solved in three steps by first obtaining an op-timal finite difference grid for discretization of the PDE,then solving a discrete EIT problem for a resistor network,and finally finding a parameterized continuous conductiv-ity. All three subproblems can be formulated as uncon-strained optimization problems, so that we can apply amodified Gauss-Newton method. Certain regularizationtechniques are considered including adaptive SVD trunca-tion of the Jacobian and a penalty function approach. Wetest our method on a well-studied case of a conductive diskin 2D for which a direct method of finding optimal gridsand resistors in a network is available. Then we generalizeour approach to a quasi-3D setting with domain of interestbeing an infinite cylinder and the conductivity which doesnot depend on Z coordinate.

Alexander MamonovRice [email protected]

MS10

On Combining Model Reduction and Gauss-

50 PD07 Abstracts

Newton Algorithms for Inverse PDE Problems

We suggest an approach to speed up the Gauss-Newton so-lution of inverse PDE problems by minimizing the numberof forward problem calls. The acceleration is based on ef-fective incorporation of the information from the previousiterations via a reduced order model (ROM). It is designedwith the help of Galerkin and pseudo-Galerkin methodsfor self-adjoint and complex symmetric problems respec-tively. The constructed ROM generates effective multivari-ate rational interpolation matching the forward olutionsand the Jacobians from the previous iterations. Numericalexamples for the inverse conductivity problem for the 3DMaxwell system show significant accelerations.

Vladimir DruskinSchlumberger-Doll [email protected]

Mikhail Y. ZaslavskySchlumberger-Doll Research,[email protected]

MS11

Galloping Instability of Viscous Shock Waves


Benjamin TexierInstitut de Mathematiques de JussieuUniversite Paris 7/Denis [email protected]

MS11

Long Tails in the Long Time Asymptotics of Quasi-Linear Hyperbolic-Parabolic Systems of Conserva-tion Laws

We consider the long-time behaviour of solutions of sys-tems of conservation laws. Liu and Zeng have given adetailed exposition of the leading order asymptotics closeto a constant background state. We extend their analy-sis by examining higher order terms in the asymptotics inthe framework of the so-called two dimensional p-system,though our methods and results should also apply to moregeneral systems. We give a constructive procedure for ob-taining these terms, and we show that their structure isdetermined by the interplay of the parabolic and hyper-bolic parts of the problem. In particular, we prove thatthe corresponding solutions develop long tails that precedethe characteristics.

Guillaume Van BaalenBoston UniversityDept. of Mathematics and [email protected]

Gene WayneBoston [email protected]

Nikola PopovicBoston UniversityCenter for BioDynamics and Department of [email protected]

MS11

Solitary Waves and Their Linear Stability inWeakly Coupled KdV Equations


Douglas WrightSchool of MathematicsUniversity of [email protected]

MS11

New Problems in Stability of Viscous ConservationLaws

Recent Lyapunov-type theorems established by Mascia-Zumbrun, Gues-Metivier-Williams-Zumbrun, Howard-Raoofi-Zumbrun, Lyng-Raoofi-Texiier-Zumbrun, Texier-Zumbrun, and others have reduced the study of nonlinearstability, asymptotic behavior, and bifurcation of viscoustraveling waves in gas dynamics combustion and MHD todetermination of point spectra of the corresponding lin-earized operator about the wave: that is, the study of theassociated eigenvalue ODE. We discuss the mathematicalissues arising in this new landscape and propose generalstrategies for their analysis, by a balanced approach com-bining numerical computation, asymptotic ODE/singularperturbation theory, and classical energy estimates.


MS12



Andrew J. ChristliebMichigan State UniverityDepartment of [email protected]

MS12

Well-Posedness of Spatially In-homogeneous Becker-Doering Kinetic System andVanishing Diffusion Limit

We prove the existence, uniqueness, and stability for theBecker-Doering infinite dimensional dynamic system ofmerging and splitting particles in the spatially inhomoge-neous case with spatial transport and diffusion. Our ap-proach and new estimates based on a modified version ofmaximum principle allow the consideration of broad classesof unbounded kinetic coefficients. The results hold bothfor parabolic, or hyperbolic, or mixed parabolic-hyperbolicsystem. Moreover, we prove that the solution with van-ishing diffusion tends to the solution of purely convectionproblem.

Pavel B. DubovskiStevens Institute of [email protected]

MS12

A Multi-Species Kinetic Model for Microelectron-ics Manufacturing Processes in the Transition and

PD07 Abstracts 51

Knudsen Regimes

We present a model for the gas flow inside chemical reactorsused in the microelectronics industry to produce computerchips. The ranges of pressures on the one and of the lengthscales of interest on the other hand necessitate the use of akinetic model for this problem. We will present the kinetictransport and reaction model developed for this problemand show results with a focus on multi-scale effects in time.

Matthias K. GobbertUniversity of Maryland, Baltimore [email protected]

MS12



Robert StrainHarvard [email protected]

MS13

Shallow Water Approximation: A MathematicalJustification.

In this talk, we will focus on a Shallow Water approxi-mation coming from the Navier-Stokes equations with freesurface. We will propose a mathematical justification ofsuch simplification based on a careful study of the approxi-mation process and based on the use of adequate estimates.This is a joint work with P. Noble (Institut Camille Jordan,Lyon 1, France).

Didier BreschUniversite de [email protected]

MS13

KdV Cnoidal Waves are Linearly Stable

Going back to considerations of Benjamin (1974), therehas been significant interest in the question of stability forthe stationary periodic solutions of the Korteweg-deVriesequation, the so-called cnoidal waves. In this paper, weexploit the squared-eigenfunction connection between thelinear stability problem and the Lax pair for the Korteweg-deVries equation to completely determine the spectrumof the linear stability problem for eigenfunctions that arebounded on the real line. We find that this spectrum isconfined to the imaginary axis, leading to the conclusionof spectral stability. An additional completeness argumentallows for a statement of linear stability.

Bernard DeconinckUniversity of [email protected]

MS13

Wall Laws at a Boundary with Non Periodic Irreg-ularity

The concern of this talk is the effect of a rough wall onan incompressible fluid. Mathematical studies of this ef-fect have been carried out under the strong assumptions ofperiodic irregularities. More realistic configurations (likea random distribution of irregularities) raise both physical

and mathematical questions. We will address some of thesequestions in the talk.

David Gerard-VaretENS [email protected]

MS13

Spectral Stability of Traveling Water Waves

The motion of the free surface of an ideal fluid under theeffects of gravity and capillarity arises in a number of prob-lems of practical interest, consequently, the reliable andaccurate numerical simulation of these ”water waves” isof central importance. Recently, in collaboration with F.Reitich, the author has developed a new, efficient, stableand high-order Boundary Perturbation scheme for the nu-merical simulation of traveling solutions of the water waveequations. In this talk we will discuss the extension of thisBoundary Perturbation technique to address the equallyimportant topic of dynamic stability of these traveling waveforms. More specifically, we describe a new numerical al-gorithm to compute the spectrum of the linearized waterwave problem as a function of the amplitude of the travel-ing wave. In addition, we will present numerical results fortwo and three dimensional waves subject to quite generalperturbations.

Dave NichollsUniversity of Illinois, Chicago,[email protected]

MS14

Regularity of the Solutions to Higher Order EllipticEquations on Non-Smooth Domains

We explore the connections between the properties of thesolution to the Dirichlet problem for a higher order ellipticequation and the geometry of the domain. In the case ofthe biharmonic equation on an arbitrary 3-dimensional do-main we prove that the gradient of the solution is bounded.The result is sharp in the sense that the solution is notnecessarily continuously differentiable. In this connection,we show that certain geometrical properties of the domainyield necessary and sufficient conditions for the continuityof the gradient. When the domain is convex, we show thatthe second derivative of the solution is bounded in all di-mensions. Moreover, we derive new results regarding thewell-posedness of the corresponding boundary value prob-lem on Lipschitz domains.

Vladimir Maz’yaThe Ohio State UniversityLinkoping University, University of [email protected]

Svitlana MayborodaMathematics, Ohio State UniversityBrown [email protected]

MS14

Regularity for Elasticity and Laplacean and FEM

We prove a well posedness result for anisotripic elasticityin weighted Sobolev spaces. Then we discuss the relevanceof this result for improving the convergence of the Finite

52 PD07 Abstracts

Element Method.

Victor Nistor, MazzucatoPennsylvania State [email protected]

MS14

Quadrature for Meshless Methods and GeneralizedFinite Element Methods

The creation of eective quadrature schemes for MeshlessMethods and Generalized Finite Element Methods is animportant problem. We discuss such quadrature schemes,and prove an energy-norm error estimate. The major hy-pothesis is that the quadrature stiness matrix has zerorow sums, a hypothesis that can be easily achieved by asimple correction of the diagonal elements. This talk isbased on joint work with Ivo Babuska, Uday Banerjee andQiaoluan(Helen) Li.

John OsbornDepartment of MathematicsUniversity Of Maryland, College [email protected]

MS14

Essential Boundary Conditions for Elliptic Equa-tions in the Generalized Finite Element Method

One of the major problems in the implementation of theGeneralized Finite Element Method is the enforcement ofessential (Dirichlet type) boundary conditions. In this talkwe address this topic in the case of boundary value prob-lems involving second-order elliptic operators. In particu-lar, we are interested in boundary data with low regularity(possibly a distribution).

Nicolae TarfuleaMathematics, Purdue University [email protected]

MS15

Transport in Porous Media. Microgemetric Effectsin Capillary Networks

We model porous media as capillary networks and developmethods to study the effect of the network geometry (i.e.“microgeometry”) on the “macroscopic” transport proper-ties of the media. In particular, we study solute transport,energy transport and fines migration and clogging.

Guillermo GoldszteinGeorgia TechAtlanta, [email protected]

MS15

Singularity of the Overall Viscous Dissipation Rateof Highly Concentrated Suspensions

We present a two-dimensional mathematical model of ahighly concentrated suspension of rigid particles in an in-compressible Newtonian fluid. The overall viscous dissipa-

tion rate W of such a suspension exhibits a singular behav-

ior. We obtain all singular terms in the asymptotics of Was an interparticle distance tends to zero. Our analysis al-lows for a complete qualitative description of microflows in

thin gaps between neighboring particles in the suspension.

Yuliya GorbTexas A&M [email protected]

Alexei NovikovPenn State [email protected]


MS15

Asymptotic and Numerical Approximation of theResonances of Thin Photonic Structures

We study the computation of the resonances of thin dielec-tric structures by using both asymptotics and PML. Ofgreat interest in the study of photonic band gap materialsis when these structures are periodic with a defect. We de-rive a limiting resonance problem as a the thickness goes tozero, and for the case of a simple resonance find a first ordercorrection. The limiting problem and correction have onedimension less, which can make the approach very efficient.Convergence estimates are proved for the asymptotics. Wethen proceed to compute these resonances by two very dif-ferent methods: the asymptotic method by solving a dense,but small, nonlinear eigenvalue problem; and PML, whichyields a large, but linear and sparse generalized eigenvalueproblem. Both methods reproduce a photonic band gaptype mode found previously by finite difference time do-main methods.

Jay GopalakrishnanUniversity of [email protected]

Shari MoskowUniversity of FloridaDepartment of [email protected]

Fadil SantosaSchool of MathematicsUniversity of [email protected]

MS15

Asymptotic Analysis of Boundary Layer Correctorsand Applications.


Daniel OnofreiMathematics DepartmentRutgers [email protected]

MS16

Mode Competition in Extended Domains

In many extended systems, governed by PDEs, the tran-sition is from a spatially uniform one-dimensional stateto a two-dimensional periodic pattern. Classically, these

PD07 Abstracts 53

systems have been simulated in periodic domains of thewavelength of the pattern. Nevertheless, this periodicitycan be broken by secondary bifurcations or in laboratoryexperiments due to imperfections. Recent computationsaccounting for spatial non-homogenities and the transitionto spatio-temporal complexity observed in hydrodynamicsystems will be discussed.

Marc AvilaUniversitad Politecnica de CatalunyaApplied [email protected]

MS16

The Role of Spatio-Temporal Symmetries in theTransition Towards Turbulence

Transitions from 2D to 3D are fundamental steps towardsturbulence. A classic problem is periodically shedding bluffbody wakes; the transition breaks spanwise O(2) symme-try. Considering only the O(2) symmetry breaking is insuf-ficient as there are additional spatio-temporal symmetrieswith important dynamic consequences; their joint actionsmust be considered. The transitions to spatio-temporalcomplexity in a variety of problems related by their sym-metry group are described.

Juan LopezArizona State UniversityDepartment of Mathematics and [email protected]

MS16

Organizing Centers and Their Connections

The competition between different instability mechanismsleads to very complicated dynamics, including homoclinicand heteroclinic phenomena. The dynamics are organizedby codimension-two bifurcations, where modes due to thedifferent instability mechanisms bifurcate simultaneously.The dynamics are explored using a three-dimensionalNavier-Stokes solver, which is also implemented in a num-ber of invariant subspaces in order to follow some unstablesolution branches and obtain a fairly complete bifurcationdiagram of the mode competitions.

Francisco MarquesUniv. Polit‘ecnica de CatalunyaApplied [email protected]

MS16

Travelling Waves in Modulated Rotating Convec-tion

Recent experiments in rotating convection have shown thatthe chaotic Kuppers-Lortz dynamics can be suppressed bysmall amplitude modulations of the rotation rate. Axisym-metric target patterns develop into axisymmetric travellingwave states as the modulation amplitude and Rayleighnumber are increased. Using fully 3D time-dependentNavier-Stokes-Boussinesq equations with physical bound-ary conditions, the experimental results are reproducednumerically gaining physical insight into the responsiblemechanism, associated with a SNIC bifurcation.

Antonio M. RubioArizona State University

[email protected]

MS17

Sharp Well-Posedness and Long-Time Bounds forthe BBM Equation

We discuss recent results on well posed-ness and long timebounds for a certain nonlinear dispersive wave equation.

Jerry BonaUniversity of Illinois at [email protected]

MS17

On Partial Regularity for the Navier-Stokes Equa-tions

A classical result of Caffarelli, Kohn, and Nirenberg statesthat the one dimensional Hausdorff measure of singularitiesof a suitable weak solution of the Navier-Stokes system iszero if the force belongs to the space L5/2+q where q >0 is arbitrary. We present a shorter proof of the partialregularity result which requires the force to belong only toL5/3+q where q > 0 is arbitrary.

Igor [email protected]

MS17

N-Vortex Dynamics on a Rotating Sphere Coupledto a Background Field

We describe a model for the evolution of N-point vorticesembedded in a background flow field that initially corre-sponds to solid-body rotation on the sphere. The full ‘em-bedded’ dynamical system is a discretization of the Eu-ler equations for incompressible flow on a rotating spheri-cal shell, hence a barotropic model of the one-layer atmo-sphere. We show how the coupling of the N-point vortices(in ring formation) to the background Rossby waves breaksthe integrability of the problem when the center of vortic-ity vector associated with the ring is misaligned with theaxis of rotation of the background field. Joint work withT. Sakajo.

Paul K. NewtonU Southern [email protected]

MS17

Time Dependent Darcy-Stokes Flow with Beavers-Joseph Interface Boundary Condition

We show that the time dependent coupled Darcy-Stokessystem is well-posed with the classical Beavers-Josephinterface boundary condition instead of the modifiedBeavers-Joseph-Saffman-Jones interface boundary condi-tion. Convergence of fully discretized finite elementschemes will be presented. Generalization to nonlinearmodels as well as applications to karst aquifer will be dis-cussed.

Xiaoming WangFlorida State Universitye-mail: TBA

54 PD07 Abstracts

Yanzhao CaoDepartment of MathematicsFlorida A and M [email protected]

Fei HuaDepartment of MathematicsFlorida State [email protected]

Max GunzburgerFlorida State UniversitySchool for Computational [email protected]

MS18

Well-Posed Boundary Value Problems for SmecticLiquid Crystals

We prove existence of minimizers to variational modelsfor smectic liquid crystals with various types of prescribedboundary values of interest for applications.

Patricia BaumanPurdue [email protected]

Daniel PhillipsDepartment of Mathematics, Purdue [email protected]

MS18

Energies of Ferroelectric Liquid Crystal Elastomersand Filaments

Models of elastomer liquid crystals are discussed and ana-lyzed, with emphasis on those presenting ferroelectric prop-erties. We discuss well-possedness of boundary value prob-lems and present an application to filament geometry.

Maria-Carme CaldererDeparment of MathematicsUniversity of Minnesota, [email protected]

MS18

Modeling of Ferroelectricity in the Chiral SmecticLiquid Crystals and Bent Core Molecules

In this presentation, we study the structure of ferroelectricliquid crystals with interactions of polarizations. Such amaterial is very interesting with respect to practical ap-plications, such as a fast switching between the active andinactive state. We introduce a model for ferroelectric liquidcrystals including nonlocal energy term and discuss equilib-rium configurations of the energy. We will focus on periodicconfigurations which appear in the physics literature.

Jinhae ParkPurdue [email protected]

MS18

Orientable and Non-Orientable Director Fields ForLiquid Crystals

Uniaxial nematic liquid crystals are often modelled using

the Oseen-Frank theory, in which the mean orientation ofthe rod-like molecules is modelled through a unit vectorfield n. This theory has the apparent drawback that it doesnot respect the head-to-tail symmetry in which n shouldbe equivalent to -n, that is, instead of n taking values inthe unit sphere S2, it should take values in the sphere withopposite points identified, i.e. in the real projective planeRP2. The de Gennes theory respects this symmetry byworking with the tensor Q=s(nn- 1/3 Id). In the case of anon-zero constant scalar order parameter s the de Gennestheory is equivalent to that of Oseen-Frank when the di-rector field is orientable. We report on a general studyof when the director fields can be oriented, described interms of the topology of the domain filled by the liquidcrystals, the boundary data and the rate of blow-up ofpossible singularities. We also analyze the circumstancesin which the non-orientable configurations are energeticallyfavoured over the orientable ones. This is joint work withJohn Ball.

John BallUniversity of Oxford, United [email protected]

Arghir ZarnescuUniversity of [email protected]

MS19



Sujeet [email protected]

MS19

Electromagnetic Transmission Spectra in PeriodicHole Arrays

Several recent experimental studies have investigated thetransmission resonances which occur at certain frequen-cies when a thin perforated conducting plate is illuminatedby an electromagnetic wave. There is hope that this phe-nomenon will prove useful in constructing filters and otherdevices for manipulating radiation in the terahertz range.We will describe an analytical and a numerical approachfor studying this problem. Numerical results will be pre-sented, along with some clues as to how the transmissionspectrum depends on the pattern and shape of the holes.

David C. DobsonUniversity of [email protected]

MS19



Alexei NovikovPenn State [email protected]

MS19

Modeling Brittle Fracture Through an Extension

PD07 Abstracts 55

of Continuum Mechanics to the Nanoscale

Described in this talk is a new approach to modeling frac-ture in brittle materials based upon an extension of con-tinuum mechanics to the nanoscale. The approach acceptsbulk continuum constitutive properties away from materialinterfaces but corrects material behavior for effects due tolong range intermolecular forces in the vicinity of materialinterfaces which in the context of fracture are the fracturesurfaces. The correction is in the form of a nonlinear mu-tual force in the bulk differential momentum balance equa-tions and a nonlinear boundary condition coming from thejump momentum balance across fracture surfaces. Bothcorrection terms derive from atom-atom interaction poten-tials. It is proved that this new modeling paradigm avoidsthe crack edge singularities found in classical elastic frac-ture theories.

Jay R. WaltonDepartment of MathematicsTexas A&[email protected]

K.-B. FuDepartment of MathematicsTexas A&M [email protected]

J. SlatteryDepartment of Aerospace EngineeringTexas A&M [email protected]

T. SendovaDepartment of MathematicsTexas A&M [email protected]

MS20

Nonlinear Stability of Time-Periodic Viscous ShockWaves

Time-period viscous Lax shocks are shown to be nonlin-early stable. The result is obtained by formulating thelinear evolution using a contour integral and using the com-ponents of the Green’s function associated to the neutraldirections to define time-dependent shifts of the phase andspatial translate. Using the integral formulation of solu-tions, we show these shifts converge to asymptotic limitsand the solution converges to the resulting space and timetranslate of the periodic shock.


Bjorn SandstedeUniversity of [email protected]

Margaret BeckUniversity of SurreyDepartment of [email protected]

MS20

Stability of Large-Amplitude Viscous Shocks

By a combination of Evans function methods, spectral en-ergy estimates, and boundary layer analysis, we are ableto prove stability of (1-D) large-amplitude viscous shocksin isentropic Navier-Stokes (or the p-system with real vis-cosity). We then describe our latest results in generalizingthis work to other systems, including the full (ideal gas)Navier-Stokes equations. Finally, we give an overview ofthe strategies used and how they may be applied moregenerally.

Jeffrey HumpherysBrigham Young [email protected]


MS20

On Darcy’s Law for Compressible Porous MediumFlows

The motion of compressible flow through porous mediumcan be modeled by compressible Euler equations with fric-tional damping. Physically, Darcy’s law is valid for largetime and the density obeys the porous medium equation.In 1-d, the mathematical verification on this conjecture wasgiven by Hsiao & Liu in 1992 for small smooth solutions.We will give a proof for any physical weak solutions.

Ronghua PanGeorgia [email protected]

MS20

Stability of Undercompressive Viscous Shock Pro-files of Hyperbolic-Parabolic Systems

We show for viscous shock profiles of arbitrary amplitudeand type, for hyperbolic-parabolic systems of conserva-tion laws, that necessary spectral (Evans function) condi-tions for linearized stability are also sufficient for linearizedand nonlinear phase-asymptotic stability, yielding detailedpointwise estimates and sharp rates of convergence in Lp,1 ≤ p ≤ ∞.


Mohammadreza RaoofiUniversity of [email protected]

MS21

Diffusive Expansion in Kinetic Theory

In this talk, the diffusive expansion to the Vlasov-Maxwell-Boltzmann system is discussed. Such an expansion yieldsa set of dissipative new macroscopic PDEs and its higherorder corrections for describing a charged fluid, where theself-consistent electromagnetic field is present. The uni-form estimate on the remainders is established via a uni-

56 PD07 Abstracts

fied nonlinear energy method and it guarantees the globalin time validity of such an expansion up to any order.

Juhi JangInstitute for Advanced [email protected]

MS21

Mathematical Problems in Reactive and Inert Ki-netic Models for Dense Gases

In the kinetic theory of simple reacting spheres (SRS), themolecules behave as if they were single mass points withtwo (or more) internal states of excitation. Collisions alterthe internal states when the kinetic energy associated withthe reactive motion exceeds the activation energy. Bothreactive and non-reactive collision events are considered tobe hard–spheres like. Thus, one considers a four compo-nent mixture A, B, A∗, B∗, and the chemical reaction ofthe type

A + B ⇀↽ A∗ + B∗.

Here, A∗ and B∗ are distinct species from A and B, andwe use the indices 1, 2, 3, and 4 for the particles A, B, A∗,and B∗ respectively. I assume that mass transfer in reac-tive collisions satisfies the condition m1 + m2 = m3 + m4,where mi denotes the mass of the i-th particle, i = 1, . . . , 4.Reactions take place when the reactive particles are sepa-rated by a distance σ12 = 1

2(d1 + d2), where di denotes the

diameter of the i-th particle.

The SRS, being a natural extension of the hard–sphere col-lisional model, reduces itself to the revised Enskog theory(RET) when the chemical reactions are turned off. Thus, itcan be considered as a good candidate for the RET analogof the reacting hard-spheres.

I will show that the kinetic theory of simple reactingspheres has an H–function. Its functional form dependson a specific type of the closure relation used in the firsthierarchy equation. In addition to its physical importance,the H–function is utilized to obtain existence and stabilitytheorems for the corresponding kinetic equations.

Jacek PolewczakCalifornia State University, [email protected]

MS21

Treatment of Multiscale Hybrid Determinis-tic/Stochastic Coupled Systems

A class of model prototype hybrid systems comprised of amicroscopic Arrhenius surface process describing adsorp-tion/desorption and/or surface diffusion of particles cou-pled to an ordinary or partial differential equation display-ing bifurcations triggered by the microscopic process is pre-sented and analyzed. These models arise from diverse ap-plications ranging from surface processes and catalysis toatmospheric and oceanic sciences.

Alexandros SopasakisUniversity of North Carolina, [email protected]

MS21

A Kinetic Formulation for Homogenization Prob-

lems

We develop an analytical tool which is adept for detectingshapes of oscillatory functions, is useful in decomposinghomogenization problems into limit-problems for kineticequations, and provides an efficient framework for the val-idation of multi-scale asymptotic expansions. We applyit first to a hyperbolic homogenization problem and trans-form it to a hyperbolic limit problem for a kinetic equation,and establish conditions determining an effective equationand counterexamples for the case that such conditions fail.Second, when the kinetic decomposition is applied to theproblem of enhanced diffusion, it leads to a diffusive limitproblem for a kinetic equation that in turn yields the ef-fective equation of enhanced diffusion.

Athanasios TzavarasUniversity of [email protected]

Pierre-Emmanuel JabinUniversity of [email protected]

MS22

Homogenization of Polyconvex Functionals

Using two-scale Young measures, I will describe the zerolevel set of an effective energy density related to a peri-odic microstructure. I will use this analysis to show thatpolyconvex energies are not closed with respect to peri-odic homogenization. The counter-example is based ona rank-one laminated structure obtained by mixing twopolyconvex functions with p-growth, where p ≥ 2 can befixed arbitrarily. I will also discuss a counter-example tothe continuity of the determinant with respect to two-scaleconvergence.

Marco BarchiesiCarnegie Mellon [email protected]

MS22

Variational Principles in L∞ and Aronsson Equa-tions

Connections between highly nonlinear degenerate PDEsarising as Aronsson equations associated to variationalprinciples for supremal functionals under general PDE con-straints will be discussed.

Marian BoceaNorth Dakota State [email protected]

MS22

Quasistatic Evolution in Plasticity with Softening

In this talk a quasistatic evolution problem in plasticitywith softening is considered in the framework of smallstrain associative elastoplasticity. The presence of a non-convex term due to the softening phenomenon requires anontrivial extension of the variational framework for rate-independent problems to the case of a nonconvex energyfunctional. In this case the use of global minimizers in thecorresponding incremental problems appears to be not jus-tified from the mechanical point of view. Thus, a differentselection criterion for the solutions of the quasistatic evo-lution problem is proposed, which is based on a viscous

PD07 Abstracts 57

approximation. This leads to a generalized formulation interms of Young measures. This is a joint work with G. DalMaso, A. De Simone, and M. Morini.

Maria Giovanna MoraSISSA, Trieste, [email protected]

MS22

Epitaxial Growth of Strained Crystalline Films: aVariational Approach

We consider strained epitaxial films grown on a relativelythick substrate in the context of plane linear elasticity.Adopting a variational approach, we look at equilibriumconfigurations; i.e., at local minimizers of the total energyfunctional, which is assumed to be the sum of the energyof the free surface of the film and the strain energy. Inthis talk we will focus on some qualitative properties ofsuch configurations that can be rigorously inferred fromthe model.

Massimiliano MoriniSISSATrieste,[email protected]

MS22

Nusselt Number Scaling in Thermal Convection

Rayleigh-Benard convection is a paradigm of nonlinear dy-namics. One key feature is the enhancement of heat trans-port as characterized by the Nusselt number, a nondi-mensional parameter whose scaling with respect to theRayleigh number has been widely studied. The backgroundmethod is a variational method for producing an upperbound on the scaling relationship. We use the backgroundmethod together with a novel background temperature pro-file and a new weighted inequality for the bi-Laplacian op-erator to produce the best analytical result to date. Thisis joint work with Felix Otto

Maria G. WestdickenbergGeorgia Institute of [email protected]

MS23

Nonlinear Water Waves in Random Bathymetry

This talk discusses the behavior of surface water waves influid domains whose boundaries are not precisely known,and therefore treated as random. It is and important topicto coastal engineering as well as to the theory of tsunamipropagation. We consider the long-wave asymptotic limitof nonlinear surface waves over bathymetry given by a sta-tionary ergodic process. It is a problem of homogenizationtheory. However it turns out that in the appropriate scal-ing regime, and in the case in which the bottom variationsare decorrolated on length scales that are long with respectto the surface waves, there remain random, realization de-pendent effects which are as important as the dispersionand nonlinearity of the problem. The talk will describethe methods of Hamiltonian averaging, the probability the-ory that goes into the analysis of these random processesand their canonical limits, and the results for surface waterwaves propagating over a random bottom. This talk shouldbe considered joint with the presentation of C. Sulem.

Walter Craig

Department of Mathematics and StatisticsMcMaster [email protected]

MS23

Solitary Waves of the Kadomtsev-PetviashviliEquation with Effects of Rotation

The rotation-modified Kadomtsev-Petviashvili (RMKP)equation was formally derived by Roger Grimshaw todescribe small-amplitude, long internal waves with weaktransverse effects in a rotating channel of shallow water,where the effects of rotation balance with weakly nonlin-ear and dispersive effects. A brief account is given of itsformulation and of connections to other model equations inoceanography and geophysics, for instance, the usual KPequation and the Ostrovsky equation. Comments are madeon the local and the global well-posedness of the Cauchyproblem for the RMKP equation. Presented with proofsare the existence of localized, traveling solutions, namely,solitary waves, of the RMKP equation and their symme-try properties, stability. Emphasis is given to the effects ofrotation.

Robin Ming ChenUniversity of [email protected]

Yue LiuUniversity of Texas at [email protected]

Vera Mikyoung HurMassachusetts Institute of [email protected]

MS23

A Shallow Water Approximation for Water Waves

In this talk we are concerned with the initial value problemfor water waves in arbitrary space dimensions. We will givea mathematically rigorous justification of the shallow waterapproximation for water waves in Sobolev spaces. Oneof the main part of the analysis is to give some uniformestimates for the Dirichlet-to-Neumann map for a scaledLaplace’s equation by using a suitable diffeomorphism.

Tatsuo IGUCHIKeio [email protected]

MS23

Water Waves Over a Random Topography

We discuss the problem of nonlinear wave motion of thefree surface of a body of fluid over a variable bottom. Theobject is to describe the character of wave propagation in along wave asymptotic regime. We assume that the bottomof the fluid region can be described by a stationary randomprocess whose variations take place on short length scales.It is a problem in homogenization theory. Our principalresult is the derivation of effective equations and a consis-tency analysis. This talk constitutes one part in a seriesof two talks on this topics, the other talk being given byWalter Craig.

Catherine Sulem

58 PD07 Abstracts

University of [email protected]

MS24

Time-Analyticity and Backwards Uniqueness forWeak Solutions of the Navier-Stokes Equations ofCompressible Flow

We prove that solutions of the Navier-Stokes equations ofthree-dimensional, compressible flow, restricted to fluid-particle trajectories, can be extended as analytic functionsof complex time. One important corollary is backwardsuniqueness: if two such solutions agree at a given time,then they must agree at all previous times as well as atsubsequent times. Additionally, analyticity yields sharpestimates for the time derivatives of arbitrary order of so-lutions along particle trajectories. Our analysis dependson a careful study of solutions of a suitably complexifiedversion of the Navier-Stokes system written in Lagrangeancoordinates.

Eugene Tsyganov, David HoffIndiana [email protected], [email protected]

MS24

Boundary Behavior of Compressible, Viscous FluidFlows

In this talk we will discuss some results on the existenceof the weak solutions to the Navier-Stokes equations forthe motion of compressible, viscous fluid flows in the half-space with the no-slip boundary condition. The emphasisis made on the boundary behavior of weak solutions withthe discontinuous density field.

Mikhail PerepelitsaVanderbilt [email protected]

MS24

From the Dynamics of Gaseous Stars to the Incom-pressible Euler Equations

A model for the dynamics of gaseous stars is introducedformulated by the Navier-Stokes-Poisson system for com-pressible, reacting gases. The combined quasineutral andinviscid limit of the NSP system in the torus T d is in-vestigated. The convergence of the Navier-Stokes-Poissonsystem to the incompressible Euler equations is proven forthe global weak solution and for the case of general initialdata.

Konstantina TrivisaUniversity of [email protected]

Donatella DonatelliUniversity of L’Aquila [email protected]

MS24

Global Solution to the Multi-DimensionalEquations of Compressible MagnetohydrodynamicFlows

An initial-boundary value problem for the three-dimensional compressible magnetohydrodynamic flow is

studied in a bounded domain with large data. The ex-istence and large-time behavior of global weak solution areestablished.

Dehua WangUniversity of PittsburghDepartment of [email protected]

MS25

Output Tracking Control of a PDE Model of a Fac-tory Production Line

Semiconductor factory production is essentially a trans-port problem that can be modelled by hyperbolic PDEs.The associated transport velocity can be modelled as astate equation leading to a non-local nonlinear equation.A major practical problem is the determination of the in-flux that generates a desired outflux over a given period oftime. Using a Lagrangian approach we present several dif-ferent algorithms to solve this problem for different typesof state equations.

Dieter ArmbrusterArizona State UniversityDept of [email protected]

Michael LaMarcaArizona State [email protected]

MS25

Optimization Techniques For Network Models II

We consider optimization problems for production net-works. This talk continues the presentation OptimizationTechniques For Network Models I. We further investigatethe optimality system. We present presolve techniques in-spired by the continuous dynamics to reduce the compu-tational effort. We show that large–scale instances of theMIP can be solved only if the new presolve techniques areused. Furthermore. we show that under certain conditionson the objective functional, the MIP is in fact equivalentto a linear programming problem. We present numericalresults for large-scale production facilities.

Simone GoettlichKaiserslautern University of Technology, [email protected]

MS25

Optimization Techniques For Network Models I

We consider optimization problems arising in the contin-uous modeling of production networks. We introduce anoptimization problem based on recently established contin-uous model for supply chains. We present an adjoint calcu-lus for solving the optimality system and discuss suitablediscretizations of this system. For particular discretiza-tions the resulting problem is in fact a MIP. We presentextensions to this model including a discretized prioritymodel as well as numerical results for sample supply chainnetworks.

Michael HertyKaiserslautern University of Technology, [email protected]

PD07 Abstracts 59

MS25

Modelling and Control of Production NetworksBased on Road Network Traffic ManagementMethods

This paper is concerned with a fluid approach towardsmodelling of production networks inspired by the macro-scopic modelling of freeway networks. The manufacturingprocess is modelled as a system whose dynamics are de-scribed by a PDE and its state is influenced by controlledinputs and uncontrolled disturbances. The modelling PDEexpresses the principle of the manufactured items’ con-servation over the range of product completion and canbe used for both simulation and control design purposes.Production systems can be controlled with the use of con-trol measures similar to those applied in freeway networks.These control measures will be reviewed and their inter-pretation within the manufacturing framework will be dis-cussed. A number of different scenarios will be examinedinvolving serial and parallel manufacturing lines.

Apostolos KotsialosDurham University, [email protected]

MS26

Gradient flow of the Aviles Giga functional

Aviles and Giga introduced the functional∫ε

2(ΔΘ)2 +

1

4ε

(1 − |gradΘ|2

)2

dx

in the context of liquid crystals. It has since appeared invarious guises for models of thin-film blistering, micromag-netics, and pattern formation. The dynamical analog (theL2 gradient flow) is also of interest, because of the impli-cations for microstructure formation. This talk considersthe singular, sharp interface limit ε → 0 which gives riseto time-evolving grain boundary networks. Formal com-putations lead to a complex free boundary problem cou-pling motion of network edges and junctions. Two junc-tion types, characterized by their geometry, are identified.The gradient flow of the (conjectured) Gamma-limit is alsoconsidered, and is show to be equivalent. Consequences ofthe resulting reduced dynamics, including dynamic scaling,will be discussed.

Karl GlasnerThe University of ArizonaDepartment of [email protected]

MS26

Singular Interfacial Energy, Faceting, and CrystalMicrostructure in Epitaxial Relaxation

The relaxation of crystal surfaces is driven by the motionof atomic line defects (“steps”). In the continuum limit,the evolution equation can be viewed as a steepest-descentsystem on the basis of a singular surface energy. Cornersingularities of the energy density correspond to macro-scopically flat regions (“facets”). In this talk I will discusscontinuum solutions consistent with the motion of steps.In particular, I will show how the crystal microstructure,in the form of individual steps on top of the physical facet,affects the slope profile macroscopically.

Dionisios MargetisUniversity of Maryland, College Park

[email protected]

MS26

On a Nonlocal Model of Biological Aggregation

I will discuss some features of a continuum model of biolog-ical aggregation introduced by Topaz, Bertozzi and Lewis.Individuals, described by the model, are attracted to eachother at a distance, but avoid overcrowding via a localdispersal mechanism. In the continuum model for the pop-ulation density the attraction is described via a nonlocaloperator, while the repulsion is modeled by a differentialoperator. We show that the density profile develops in-terfaces between a near-constant-density aggregate stateand the empty space. The interfaces evolve under surface-tension-like “forces”. More precisely, using the gradientflow structure and asymptotic analysis, we show that thesharp interface limit for the interfacial motion is the Hele-Shaw flow. On long time scales the interfacial motion leadsto coarsening of length scales present in the system. Therate of coarsening can be investigated using the Kohn-Ottoframework.We will describe a geometric viewpoint, whichunites the coarsening results in a variety of interfacial mod-els. The talk is based on joint work with Andrea Bertozzi.

Dejan SlepcevCarnegie Mellon [email protected]

MS26


I will discuss some asymptotic limits of the Ginzburg-Landau and Chern-Simons-Higgs energies via Gamma con-vergence methods.

Daniel SpirnUniversity of [email protected]

MS26

The IVP for Euler/Euler-Poisson Type Systems

The ”sticky particles” model at the discrete level is em-ployed to obtain global solutions for a class of systems ofconservation laws among which lie the pressureless Eulerand pressureless Euler-Poisson systems. We consider thecase of finite, nonnegative initial Borel measures with fi-nite second-order moments, along with continuous initialvelocities of at most quadratic growth and finite energy.

Adrian TudorascuGeorgia Institute of [email protected]

MS27

On a Diffuse Interface Model for a Binary Mixtureof Compressible Viscous Fluids

We consider a model describing the time evolution of a bi-nary mixture of compressible viscous fluid consisting of thecompressible Navier-Stokes system coupled with a Cahn-Hilliard equation for the order parameter. We show thatthe problem posesses a global-in-time weak solution for anyfinite energy initial data. Related questions are also dis-cussed.

Eduard Feireisl

60 PD07 Abstracts

Mathematical Institute ASCR, Zitna 25, 115 67 Praha 1Czech [email protected]

Helmut AbelsMax Planck [email protected]

MS27

Contact Discontinuity for Jin-Xin Relaxation Sys-tem

We construct the contact waves for Jin-Xin relaxation sys-tem in 1-D. Such waves are relaxation correspondence ofcontact discontinuity for hyperbolic conservation laws atequilibrium. We further prove that they are stable undergeneric small intial perturbations.

Ronghua PanGeorgia [email protected]

MS27

Stability of Perfect-Fluid Shock Waves in Specialand General Relativity

For general relativity, the persistence problem of shockfronts in perfect fluids is also a continuation problem fora pseudo-Riemannian metric of reduced regularity. Wesolve the problem by considerations on a Cauchy problemwhich combines a well-known formulation of the Einstein-Euler equations as a first-order symmetric hyperbolic sys-tem and Rankine-Hugoniot type jump conditions for thefluid variables with an extra (non-)jump condition for thefirst derivatives of the metric. As in non-relativistic set-tings, the shock front must satisfy a Kreiss-Lopatinski con-dition in order for the persistence result to apply.

Mohammadreza RaoofiUniversity of Minnesota, School of [email protected]

Heinrich FreistuehlerUniversit”at [email protected]

MS27



Zhouping XinChinese University of Hong [email protected]

MS28

On a Free Boundary Problem Arising from Mate-rial Science

In this talk, we will study a free boundary problem arisingfrom material science. The regularity of both the phasefunction and the free boundary will be discussed.

Huiqiang JiangDepartment of Mathematics, University of [email protected]

MS28

Threshold-Based Quasi-Static Brittle Damage Evo-lution

We will first consider a variational model for damage pro-posed by Francfort and Marigo, and then give an existenceresult (for a quasi-static version) by Francfort and Gar-roni. A case for a strain-threshold damage model will bemade, and precise definitions for both an unrelaxed (i.e.,the damage region is a set at each time) and relaxed (i.e.,the damage ”set” is given by a density function, togetherwith an effective tensor that stores the microstructure ofthe damage) quasi-static evolution will be given. It turnsout that solutions of the unrelaxed variational problem aresolutions to the unrelaxed threshold problem, but there isan interesting issue in showing the same for the relaxedproblems. This issue is related to a question regarding G-closure and the relative scalings of damage microstructureat different times. A new variational formulation, arguablymore physical than the existing one, will be given, and wewill show that relaxed solutions of this variational prob-lem are solutions of the relaxed threshold problem. This isbased on joint work with A. Garroni.

Christopher LarsenWorcester Polytechnic InstituteMath [email protected]

MS28

Nonuniqueness in a Free Boundary Problem FromCombustion

We will present a rigorous justification for a nonuniquenessexample in the parabolic free boundary problem with fixedgradient condition, which appears in combustion theory todescribe the propagation of premixed equidiffusional flamesin the limit of high activation energy.

Aaron Yip, Arshak PetrosyanPurdue UniversityDepartment of [email protected], [email protected]

MS28

Infinity Laplace Equation And Related Issues

The speaker plans to introduce the basic theory of Infin-ity Laplace Equation and related topics. In particular, heplans to cover some recent progress in the theory.

Peiyong WangDepartment of MathematicsWayne State [email protected]

MS28

Global Minimizers of Chern-Simons-Higgs Func-tional Below Critical Magnetic Field

For CSH functional of superconductors in an external field,we prove the global minimizer is vorticesless when the ex-ternal field is below critical field. It is a joint work withDaniel Spirn from Minnesota

Xiaodong YanMichigan State UniversityDepartment Of [email protected]

PD07 Abstracts 61

MS29

An Analytic Model in Finding Geodesic Path Be-tween Shapes

Curve matching plays a crucial role for many problems incomputer vision and medical image analysis. In this paper,we built a novel model based on Hellinger distance for curvematching. In this model, we can construct a geodesic pathbetween two shapes analytically. Besides, this model withslight modifications is capable of finding optimal correspo-nendents among a group of curves simultaneously. Ourexperiments show the effectiveness of this model.

Pengwen Chen, Yunmei ChenUniversity of [email protected], [email protected]

MS29

Upper Bounds on the Coarsening Rate of Discrete,Ill-Posed, Nonlinear Diffusion Equations

We prove a weak upper bound on the coarsening rate of thediscrete-in-space version of an ill-posed, nonlinear diffusionequation. The continuum version of the equation violatesparabolicity and lacks a complete well-posedness theory. Inparticular, numerical simulations indicate sensitive depen-dence on initial data. Nevertheless, models based on itsdiscrete-in-space version, which we study, are widely usedin a number of applications, including population dynamics(chemotactic movement of bacteria), granular flow (forma-tion of shear bands), and computer vision (image denoisingand segmentation). Our bounds have implications for allthree applications. They are obtained by following a recenttechnique of Kohn and Otto. Based on joint works withJohn Greer and Dejan Slepcev.

Selim EsedogluDepartment of MathematicsUniversity of [email protected]

MS29

A Second Order Variational Model for StaircaseReduction

The one-dimensional version of the higher order totalvariation-based model for image restoration proposed byChan, Marquina, and Mulet is analyzed. A suitable func-tional framework is proposed in which the minimizationproblem is well posed, and it is proved analytically thatthe higher order regularizing term prevents the occurrenceof the staircasing effect.

Irene FonsecaCarnegie Mellon [email protected]

MS29

Nonlocal Operators for Image Processing

We propose the use of nonlocal operators to define newtypes of flows and functionals for image processing. A mainadvantage over standard algorithms is the ability to simul-taneously the texture and cartoon parts of images. Ap-plications to denoising, segmentation and inpainting aregiven. This work builds on much research done earlier ina host of related fields.

Guy Gilboa

UCLA [email protected]

Stanley J. OsherUniversity of CaliforniaDepartment of [email protected]

MS29

Redistancing by Flow of Time Dependent EikonalEquation

Construction of signed distance to a given inteface is atopic of special interest to level set methods. There arecurrently, however, few algorithms that can efficiently pro-duce highly accurate solutions. We introduce an algorithmfor constructing an approximate signed distance functionthrough manipulation of values calculated from flow of timedependent eikonal equations. We provide operation countsand experimental results to show that this algorithm canefficiently generate solutions with a high order of accuracy.Comparison with the standard level set reinitialization al-gorithm shows ours is superior in terms of predictabilityand local construction,critical, for example, in local levelset methods. We further apply the same ideas to extensionof values off interfaces. Together, our proposed approachescan be used to advance the level set method for fast andaccurate computations of the latest scientific problems.

L.T. ChengUniversity of California, San [email protected]

Richard TsaiDepartment of MathematicsUnivesity of [email protected]

MS30

A Long-Short Wave Equation in Fermi-Pasta-UlamLattice

It is well known that the Boussinesq equation, further, theKdV equation can be derived from the Fermi-Pasta-Ulam(FPU) lattice as a long wave continuum approximation.Taking into account both the acoustic and optical modes,we derive a coupled partial differential equations in attemptto study the long-short wave interaction in FPU lattice.Some numerical and analytical results of this long-shortwave equation will also be presented.

Baofeng FengU. of Texas-Pan [email protected]

MS30

Dynamics of the Localized Waves in Rayleigh-Benard Convection With Heterogeneous Concen-tration Field

We study the Rayleigh-Benard convection in binary fluidmixture by an amplitude equation derived by Riecke. Thisequation has a variable, corresponding to the concentra-tion field, which works as a heterogeneous driving force forfluid motion. This equation gives interesting dynamicalphenomena among localized waves(pulses), i.e., formationof bound states by both co- and counter-propagating pulsesand various input-output relation by pulse collision. These

62 PD07 Abstracts

phenomena are analyzed in terms of the network topol-ogy constructed by stable and unstable manifolds in phasespace, forcusing on the role of the concentration field.

Yasumasa Nishiura, Makoto IimaResearch Institute for Electronic ScienceHokkaido [email protected],[email protected]

MS30

Pinning and Depinning for Particle-Like Solutionsin Dissipative Systems

Particle-like dissipative patterns arise in many fields suchas chemical reaction, gas-discharge system, liquid crystal,binary convection, and morphogenesis. We discuss aboutthe dynamics of moving particles in heterogeneous media,especially we are interested in the case in which travelingpulses or spots encounter heterogeneities of bump type.A variety of dynamics are produced such as penetration,annihilation, rebound, splitting, and relaxing to an orderedpattern, i.e., pinning. It turns out that global bifurcationsuch as heteroclinic one and a basin switching controlledby the unstable manifolds of saddless called scattors playa crucial role for the transition from pinning to depinning.We also discuss a wave-generator created near the jump ofheterogeneity.

Yasumasa NishiuraRIES, Hokkaido [email protected]

Takashi TeramotoChitose Institute of Science and [email protected]

MS30

Behavior of an Amoeba Crossing an EnvironmentalBarrier

In this talk, we will focus on the effect of inhomogeneousenvironments on tip migration behavior in the Physarumplasmodium of true slime mold. The plasmodium migrat-ing in a narrow lane shows penetration, rebound, and split-ting dynamics when it encounters the presence of a chem-ical repellent, quinine. In order to understand the dynam-ics, a phenomenological model based on reaction-diffusionsystems is proposed. We discuss how the origin of the threedifferent outputs could be reduced to the hidden instabili-ties of internal dynamics of the tip.

Kei-Ichi UedaKyoto University, Japanueda [email protected]

MS31

Coupling of Finite Element and Boundary IntegralMethods for a Scattering Problem in Near-FieldOptics

This talk is concerned with the solution of the Helmholtzequation in the modeling of a scattering problem for one ofimportant experimental modes of near-field optics, photonscanning tunneling microscopy. The well-posedness for thecontinuous and discrete problems of coupling finite elementand boundary integral methods will be addressed, error

estimates for the coupling procedure will be provided, andnumerical examples will be presented.

Gang BaoMichigan State [email protected]

Peijun LiDepartment of MathematicsUniversity of [email protected]

MS31

Superposition of Multi-Valued Solutions in HighFrequency Wave Dynamics

The weakly coupled WKB system captures high frequencywave dynamics in many applications. For such a systema level set method framework has been recently developedto compute multi-valued solutions to the Hamilton-Jacobiequation and evaluate position density accordingly. In thispaper we propose two approaches for computing multi-valued quantities related to density, momentum as well asthe energy. Within this level set framework we show thatphysical observables evaluated in [?, ?] are simply the su-perposition of their multi-valued correspondents. A seriesof numerical tests is performed to compute multi-valuedquantities and validate the established superposition prop-erties.

Hailiang LiuIowa State [email protected]

MS31

A Field Space-Based Level Set Method for Com-puting Multi-Valued Solutionsto 1D Euler-PoissonEquations

We present a field space based level set method for com-puting multi-valued solutions to one-dimensional Euler-Poisson equations. The system of these equations hasmany applications, and in particular arises in semiclassicalapproximations of the Schrodinger-Poisson equation. Theproposed approach involves an implicit Eulerian formula-tion in an augmented space — called field space, whichincorporates both velocity and electric fields into the con-figuration. Both velocity and electric fields are capturedthrough common zeros of two level set functions, whichare governed by a field transport equation. Simultane-ously we obtain a weighted density f by solving again thefield transport equation but with initial density as startingdata. The averaged density is then resolved by the inte-gration of the obtained f against the Dirac delta-functionof two level set functions in the field space. Moreover,we prove that such obtained averaged density is simply alinear superposition of all multi-valued densities; and theaveraged field quantities are weighted superposition of cor-responding multi-valued ones. Computational results arepresented and compared with some exact solutions whichdemonstrate the effectiveness of the proposed method.

Zhongming WangIowa State [email protected]

PD07 Abstracts 63

MS31

Computation of the Radiative Transfer Equationwith Interface Reflection and Transmission

Abstract not available at time of publication

Xu YangDepartment of MathematicsUniversity of [email protected]

MS32

Variational Principles for Supremal Functionalsand Applications to Polycrystal Plasticity

Γ-convergence results for a general class of power-law func-tionals acting on divergence free fields are presented. Someconsequences are described in the settings of antiplaneshear and plane stress plasticity.

Marian BoceaNorth Dakota State [email protected]

MS32

Inverse Homogenization - Recovery of the Struc-ture of Composite Materials

The talk deals with inverse homogenization problem whichis a problem of deriving information about the microgeome-try of composite material from its effective properties. Theapproach is based on reconstruction of the spectral mea-sure in the analytic Stieltjes representation of the effectivetensor of two-component composite. This representationrelates the n-point correlation functions of the microstruc-ture to the moments of the spectral measure, which con-tains all information about the microgeometry. The prob-lem of identification of the spectral function from effectivemeasurements in an interval of frequency has a unique so-lution, however the problem is ill-posed. The talk discussesseveral stabilization techniques as well as Pade approxima-tion used to reconstruct the spectral function. Results ofreconstruction of microstructural parameters are shown forvisco-elastic composites and for composites of two materi-als with different complex permittivity. The reconstructedspectral function can be used to compute other effectiveproperties of the same composite; this gives solution tothe problem of coupling of different effective propertiesof a two-component composite material with random mi-crostructure.

Elena CherkaevUniversity of UtahDepartment of [email protected]

MS32

Coupled-Mode Theory of Wave Resonance in a Pe-riodic Medium

We study the Bragg resonance of surface water waves by atwo-dimensional array of vertical cylinders covering a largearea of the sea. Employing some concepts of solid state andcrystallography, we combine the techniques homogeniza-tion and inner-outer expansion to derive two-dimensionalequations for the envelopes of scattered waves resonated bya plane incident wave. Solutions for two- and three-wavescattering are obtained for a long strip of cylinder array.

Roles of the bandgaps are examined.

Chiang C. MeiDept of Civil and Environmental [email protected]

Yile LiShell International Exploration and Production IncHouston TXyile.li@gmailcom

MS32

G-Convergence and Homogenization of Fluid-Structure Interaction and Two-Fluid Flow

We study two homogenization problems for flows with mov-ing interfaces: a small deformation model of fluid-structureinteraction and a Stokes flow of two immiscible fluids with-out surface tension. Since the interfaces are moving, wework in the framework of G-convergence that requires noa priori geometric assumptions. To pass to the limit wepropose a new construction of divergence-free oscillatingtest functions. The construction involves several auxiliaryfuncitons that are found by solving auxiliary problems.The effective equations for fluid-structure interaction areof viscoelastic type with long memory terms not present inthe microscale problems. In the case of two fluid flow, theeffective equations are of the same type as microscale prob-lems, but the effective viscosity tensor depends of time ingeneral. Since constitutive equations for viscosity shouldnot depend on time explicitly, it would be desirable to ob-tain sufficient conditions for existence of an equation ofstate that would prescribe the effective viscosity as a time-independent function of the effective density and effectivepressure. We present some results in this direction (jointwork with Elena Cherkaev).

Alexander PanchenkoWashington State [email protected]

MS33

High Confinement Limit and Time Averaging forthe Gross-Pitaevski Equation


Naoufel Ben AbdallahUniversity of [email protected]

MS33

Quantum Charge Transport in Random Media

In this talk, we discuss some recent results addressing quan-tum charge transport in a weak random potential. Wefocus on the derivation of Boltzmann equations, obtainedfrom a macroscopic, hydrodynamic limit, both in the one-particle, and in the fermionic manybody framework. Inconnection with the latter, we report on a recent joint workwith I. Sasaki.

Thomas ChenPrinceton [email protected]

64 PD07 Abstracts

MS33

5-Moment-Corrections to a Drift-Collision BalanceModel of Quantum Transport

We present a fluid–dynamical model which describesquantum–driven electron transport in the so-called “drift–collision balance” regime. Here, external field and inter-action with a phonon bath are the dominant phenom-ena and have comparable strength. More precisely, inthe rescaled Wigner–BGK equation they appear at thesame order in the parameter ε. The BGK term containsthe (Wigner) O(〈∈)–correction to the thermal equilibriumfunction, parametrized by the electron position density andthe bath temperature, which models the state to whichthe system shall be driven by the sole interaction with thecrystal ions. Via a Chapman–Enskog procedure we findO(ε)–correction to the solution of the scaled Wigner-BGKequation. Then, by applying the moment method, we findO(ε)–corrections to the equations for position density, ve-locity flux and energy density. In these corrective terms ap-pear five moments of the O(∞)–in–ε solution and they areof diffusive type in the fluid unknowns. High-field O(〈∈)–corrections are present that are peculiar of quantum-driventransport.

Giovanni FrosaliDept. of Applied MathematicsUniversity of [email protected]

Giovanni BorgioliDept. of ElectronicsUniversity of [email protected]

Chiara ManziniDept. of Applied Mathematics ”G.Sansone”Universita’ di Firenze, [email protected]

MS33

On the Global Existence of Smooth Solution to the2-D FENE Model

In two space dimension, we prove the global existenceof smooth solutions to a coupled microscopic-macroscopicFENE dumbbell model which arises from the kinetic theoryof diluted solutions of polymeric liquids with noninteract-ing polymer chains.

Ping ZhangBeijing Academy of [email protected]

MS34

A Novel Algorithm for Fluid-Structrure Interac-tion in Blood Flow

The fluid-structure interaction in blood flow involves twomedia, blood and arterial walls, whose densities are of thesame order of magnitude. This leads to a highly nonlin-ear interfacial coupling. To date, only strongly coupledalgorithms (implicit/monolithic) seem applicable to bloodflow simulations. In this talk, we present a novel classof loosely coupled (partitioned) algorithms which combinethe stability properties of implicit schemes with the lowcomputational costs of explicit schemes.

Suncica Canic

Department of MathematicsUniversity of [email protected]

Tsorng-Whay PanDepartment of MathematicsUniversity of [email protected]

Roland GlowinskiUniversity of [email protected]

Giovanna GuidoboniUniversity of HoustonDepartment of [email protected]

Sergey LapinDepartment of MathematicsWashington State [email protected]

MS34

Adhesion and Slough-off of Chondrocytes UnderControlled Flow Conditions

Cell adhesion under conditions of fluid flow plays an impor-tant role in many physiological and biotechnological pro-cesses. Recently auricular chondrocytes were studied forlining a stent, which is a potential treatment of coronaryartery disease. To improve the biocompatibility, it is cru-cial for cells to remain firmly on surface initially. In thispresentation, we mathematically model and simulate adhe-sion and slough-off of chondrocytes in shear flow. We firstdiscuss the Adhesive Dynamics for bonding between celland surface; then introduce a Lagrange multiplier basedfictitious domain method to solve the fluid-cell system. Nu-merical results for the cells in shear flow and the case withdeterministic adhesion will be presented.

Jian HaoUniversity of HoustonDepartment of [email protected]

MS34

Critical Thresholds in a Quasilinear HyperbolicModel of Blood Flow

We study critical threshold phenomena in a quasilinear hy-perbolic model of blood flow through compliant axisym-metric vessels with viscous effects. We identify criticalthresholds for global existence of smooth solutions and forthe finite time singularities. We prove the existence of asmooth solution under some conditions on the slope of thepressure data. We obtain that for certain physiologicallyrelevant data one has shock formation.

Tong LiDepartment of MathematicsUniversity of [email protected]

Suncica CanicDepartment of MathematicsUniversity of [email protected]

PD07 Abstracts 65

MS34

Simulating Blood Flow and Vessel Dynamics inPatient-Specific Arterial Models

In recent years, image-based computational methods havebeen described for modeling blood flow in the cardiovascu-lar system and test hypotheses related to the localizationof vascular disease. We will present results related to mod-eling blood flow and vessel dynamics in the systemic andpulmonary circulations of individual subjects. We then de-scribe a new approach for cardiovascular treatment plan-ning in which the physician utilizes computational tools toconstruct and evaluate a combined anatomic/physiologicmodel to predict the outcome of alternate treatment plansfor an individual patient. In order to ensure that thesepredictive tools are clinically relevant, they must faith-fully represent the anatomy and physiology of individualpatients. Recent progress in constructing subject spe-cific anatomic models from medical imaging data, acquir-ing physiologic data under resting and exercise conditions,planning treatments, and modeling blood flow and vesseldeformation will be described. Verification of FSI methodswith analytical solutions will be discussed. Results fromin vitro and animal studies utilizing magnetic resonanceimaging techniques to validate computational blood flowsimulations will be presented.

Charley TaylorStanford UniversityMechanical [email protected]

MS35

Maximum Enstrophy Production in the 3D Navier-Stokes

It is still not known whether solutions to the 3D Navier-Stokes equations for incompressible flows in a finite peri-odic box can become singular in finite time. It is knownthat solutions remain smooth as long as their enstrophy(mean-square vorticity) is finite. The generation rate of en-strophy is given by a functional that can be bounded usingelementary functional estimates. Those estimates establishshort-time regularity but do not rule out finite-time singu-larities in the solutions. In this work we formulate andsolve the variational problem for the maximal growth rateof enstrophy and display flows that generate enstrophy atthe greatest possible rate. Implications for questions of reg-ularity or singularity in solutions of the 3D Navier-Stokesequations are discussed.

Charles R. DoeringUniversity of MichiganDept of [email protected]

Lu LuWachovia InvestmentsNew York, [email protected]

MS35

The Bifurcation and Stability Analysis of Double-Diffusive Convection

In this talk, we present a bifurcation and stability analy-sis on the double-diffusive convection. Both steady state

bifurcation and the Hopf bifurcation are considered.

Tian MaSichuan University/

Shouhong WangDepartment of MathematicsIndiana [email protected]


Chun-Hsiung HsiaU Illinois [email protected]

MS35

Stochastic Parametrization of Wave Breaking andWave/Current Interactions

If wave breaking modifies the Lagrangian fluid orbitalpaths by inducing an uncertainty in the path itself andthis uncertainty on wave motion time scales is observableas additive noise, it is shown that within the context ofa wave/current interaction model for basin and shelf scalemotions it persists on long time scales. A model of con-servative dynamics, developed in collaboration with Laneand McWilliams, provides the general framework for thedynamics of the wave-current interactions. In addition tothe deterministic part, the vortex force, which couples thetotal flow vorticity to the residual flow due to the waves,will have a part which is associated with the dissipativemechanism. At the same time the wave field will expe-rience dissipation, and tracer advection is affected by theappearance of a dissipative term in the Stokes drift veloc-ity. Consistency leads to other dynamic consequences: theboundary conditions are modified to take into account thediffussive process and proper mass/momentum balances atthe surface of the ocean. I will present an overview ofthe model, how wave dissipation effects are included as astochastic parametrization, and show preliminary resultson a model problem.

Juan M. RestrepoUniversity of [email protected]

MS35

On the Motion and Regularity of Vortex Sheetswith Surface Tension

We prove well-posedness of vortex sheets with surface ten-sion in the 3D incompressible Euler equations with vor-ticity. This is a moving boundary-value problem for themotion of two incompressible perfect fluids separated by asurface of discontinuity, wherein the motion of the fluid in-teracts with the motion and regularity of the vortex sheetat highest-order.

Steve ShkollerUniviversity of California, DavisDepartment of [email protected]

66 PD07 Abstracts

MS36

Incompressible Fluids in Thin Domains withNavier Friction Boundary Conditions

This study is motivated by problems arising in oceanicdynamics. Our focus is the Navier-Stokes equations in athree-dimensional domain Ωε, whose thickness is of order0(ε) as ε → 0, having non-trivial topography. The velocityvector field is subject to the Navier friction boundary con-ditions on the bottom and the top boundaries of Ωε withthe friction coefficients γ0,ε and γ1,ε, respectively. Assumethat γ0,ε and γ1,ε are of order 0(1), and |γ0,ε − γ1,ε| is oforder 0(3/4) as ε → 0. It is shown that if the initial veloc-ity field and the body force belong to ”large sets”, then thestrong solution of the Navier-Stokes equations exists for alltime. Our proofs rely on the study of the dependence of theStokes operator on ε, and the non-linear estimate in whichthe contributions of the boundary integrals are non-trivial.

Luan T. HoangSCHOOL OF MATHEMATICSUNIVERSITY OF [email protected]

MS36

Averaged Bounds on Moments for the 2D Navier-Stokes Equation

Upper and lower bounds for the ensemble averages of en-ergy, enstrophy, and palinstrophy (the first three momentsof the 2D, periodic, Navier-Stokes equations) are derivedboth in the general case, and in the case where the en-ergy spectrum for fully developed turbulence holds. In theturbulent case, the bounds are sharp, up to a logarithm,and provide a new lower bound on the Landau-Lifschitzdegrees of freedom. A key Sobolev-type estimate of theinertial term is shown to be sharp on a significant portionof the global attractor.

Michael S. JollyIndiana UniversityDepartment of [email protected]

Ciprian FoiasTexas A & M [email protected]

Radu DascaliucIndiana UniversityDepartment of [email protected]

MS36

Discrete in Time Coupling and Synchronization inthe 2D Navier-Stokes Equations


Eric OlsonUNRejolson

MS36

Singularly Perturbed Convection-Diffusion Equa-tions with a Turning Point

Turning points occur in many circumstances in fluid me-

chanics. When the viscosity is small, very complex phe-nomena can occur near turning points, which are not yetwell understood. A model problem, corresponding to a lin-ear convection-diffusion equation (e.g., suitable lineariza-tion of the Navier-Stokes or Benard convection) equationis considered. Our analysis shows the diversity and com-plexity of behaviors and boundary or interior layers whichalready appear for our equations simpler than the Navier-Stokes or Benard convection equations. Of course the di-versity and complexity of these structures will have to betaken into consideration for the study of the nonlinearproblems. In our case, at this stage, the full theoreticalasymptotic analysis is provided.

Chang-yeol JungIndiana [email protected]

Roger TemamInst. f. Scientific Comput. and Appl. Math.Indiana [email protected]

MS37

Thin-Film Coarsening Driven by Surface Relax-ation and the Ehrlich-Schwoebel Effect

The surface of an epitaxially growing thin film often ex-hibits a mound-like structure with its characteristic lateralsize increasing in time. In many material systems, sucha coarsening process is driven mainly by two competingmechanisms. One is the surface relaxation described byhigh-order gradients of the surface profile, and the otherthe Ehrlich-Schwoebel effect which is the upper-lower ter-race asymmetry in the adatom attachment and detachmentto and from atomic steps. In this talk, we present an analy-sis of a class of partial differential equation models that aremathematically gradient-flows of some effective free-energyfunctionals describing these mechanisms. Our analysis con-sists of two parts: (1) variational properties of the energies,such as “ground states” and their large-system-size asymp-totics, showing the unboundedness of surface slope and re-vealing the relation between some of the models; (2) rigor-ous bounds for the scaling law of the roughness, the rate ofincrease of surface slope, and the rate of energy dissipation,all of which characterize the coarsening process.

Bo LiDepartment of Mathematics, UC San [email protected]

MS37

Modeling and Analysis of Stepped Crystal Surfaces

The morphological evolution of crystal surfaces is stud-ied as a prototypical case of modeling across the scales.At the nanoscale, the governing discrete schemes describethe motion of line defects (“steps”) of atomic height. Atthe macroscale, the resulting free-boundary problems forPDEs retain a novel microstructure. I will focus on re-cent progress and challenges in answering the question: Forwhat continuum theory is step motion a consistent numer-ical solution scheme ? I will present the continuum lawsand free-boundary problems that result from step motion.I will also explore the validity of the continuum theory fromthe perspective of kinetic BBGKY-type hierarchies for stepcorrelation functions.

Dionisios Margetis

PD07 Abstracts 67

University of Maryland, College [email protected]

MS37

Epitaxial Growth, Atomistic Simulations on Diffu-sive Time Scales

We apply a phase field crystal model to epitaxial growthand study equilibrium shapes of islands and mounds. Themodel is derived from classical density functional theory(DFT) and leads to a higher order parabolic PDE for theatom density. The derivation from DFT as well as a coarsegraining which leads to classical phase field models will bediscussed. Furthermore efficient numerical methods basedon finite element discretizatioins will be shown.

Andreas RaetzUniversity of [email protected]

MS37

A Continuum Model for the Long-Range ElasticEffect on Epitaxial Surfaces

Below the roughening transition, the surface of an epitaxialfilm consists of steps whose distribution and motion deter-mine the surface morphology. The elastic effects on theepitaxial surface provide a driving force for the motion ofthese surface steps. The stress in the epitaxial film, whichmay be due to the misfit between the lattice constants inthe film and in the substrate, generates a long-range elas-tic effect on the epitaxial surface. We present a continuummodel for this long-range elastic effect, incorporating thediscrete features of the stepped surface. The model is ob-tained by taking continuum limit of the discrete model forthe elastic interaction between surface steps.

Yang XiangHong Kong University of Science and [email protected]

MS38

A Variational Theory for Point Defects in Patterns

We derive a rigorous scaling law for minimizers in a naturalversion of the regularized Cross-Newell model for patternformation far from threshold. These energy-minimizing so-lutions support defects having the same character as whatis seen in experimental studies of the corresponding physi-cal systems and in numerical simulations of the microscopicequations that describe these systems.

Shankar C. VenkataramaniUniversity of ArizonaDepartment of [email protected]

Nick ErcolaniUniversity of [email protected]

MS38

Scaling Dynamics for Smoluchowski’s CoagulationEquations with Dust and Gel

We study limiting behavior of rescaled size distributionsthat evolve by Smoluchowski’s rate equations for coagula-tion, with rate kernel K=2, x+y or xy. We find that the

dynamics naturally extend to probability distributions onthe half-line with zero and infinity appended, represent-ing populations of clusters of zero and infinite size. The“scaling attractor” (set of subsequential limits) is compactand has a Levy-Khintchine-type representation that lin-earizes the dynamics and allows one to establish severalsignatures of chaos. In particular, for any given solutiontrajectory, there is a dense family of initial distributions(with the same initial tail) that yield scaling trajectoriesthat shadow the given one for all time.

Govind MenonApplied Mathematics, Brown [email protected]

Robert PegoCarnegie Mellon [email protected]

MS38

Aggregation of Floating Particles by Surface Ten-sion Forces: From Nano- to Millimeter Scales.

We derive an evolution equations for self-assembly of float-ing particles. First, we present results of numerical simu-lations showing that energy density of a realistic particleclump of an arbitrary shape is described by a surprisinglysimple formula, and that the continuum approximationmay be used for clumps with as few as five particles. Next,we deduce an evolution equation for density evolution forin the case of central potential and describe analyticallystationary and spatio-temporal solutions forming the baseof the dynamics. Finally, we suggest an evolution equationsfor particles with interaction potential dependent not onlyon their density, but also on the mutual orientation. In theprocess, we also derive the analogue of Darcy’s law and cor-responding equation of motion for an arbitrary geometricquantity.

Darryl HolmImperial College London andLos Alamos National [email protected]

Vakhtang PutkaradzeColorado State [email protected]

MS38

Rigorous Scaling Laws and Crossover Behavior inElastic Ridges


Shankar C. VenkataramaniUniversity of ArizonaDepartment of [email protected]

MS39

Controllability of a Cochlea and Related Fluid-Elastic Systems

Standard 2-dimensional cochlea model consists of one-dimensional elastic structure surrounded by incompressiblefluid within cochlear cavity. Dynamics are typically drivenby pressure differential across the basilar membrane trans-mitted through round and oval windows. Idealized basilar

68 PD07 Abstracts

membrane is modeled as infinite array of oscillators andfluid is described by Laplace’s equation. We show approx-imate controllability with control acting on arbitrary openset of membrane. If membrane has longitudinal elasticity,then exact controllability can be proved.

Scott HansenIowa State UniversityDepartment of [email protected]

MS39

Optimal Static Control of Groundwater

Control theory has been less developed for elliptic par-tial differential equations than it has for hyperbolic andparabolic variety. In this talk we indicate how control canbe applied to study certain problems for elliptic systems.We use Darcy’s equation for groundwater flow togetherwith mathematically idealized representation of control in-tervention by means of pumping and transport as an ex-ample. We will indicate the relevant theoretical questionsand present the results of some computational simulations.

David RussellVirginia [email protected]

MS39

Multiscale Modeling of Preferential Flow

We discuss new multi-scale models of fully saturated flowcoupled to transport through highly heterogeneous media.These are systems of partial differential equations that in-clude the usual diffusion effects from various spatial scalesand additionally model the effects of local advective trans-port through fast flow regions.

Malgorzata Peszynska, Ralph ShowalterDepartment of MathematicsOregon State [email protected],[email protected]

MS39

Fluid-Structure Interaction Problems

We consider parabolic-hyperbolic fluid-structure interac-tion PDE model (dimensions 2 and 3): structure of arbi-trary shape, modeled by the system of dynamic elasticity,is immersed in a fluid modeled by Navier-Stokes equation,with coupling at the interface. In linear case, semigroupwell-posedness in the energy space is asserted. Spectralproperties of the generator are described. Strong stabilityis analyzed. To obtain uniform stability, the interface con-dition is modified to become dissipative. Backward unique-ness result is presented. uniqueness.

George AvalosUniversity of Nebraska-LincolnDepartment of Mathematics and [email protected]

Roberto TriggianiUniversity of [email protected]

MS40

On the Spectrum Associated with Periodic Wavesfor Generalized KdV Equations


Todd KapitulaDepartment of MathematicsCalvin [email protected]

MS40

Semi-Strong N-Pulse Interaction in a Hypberbolic-Parabolic System

We consider the semi-strong interaction of optical pulsesin a thermally detuned optical parametric oscillator. Thepulses evolve under a parametrically forced nonlinearSchrodinger equation coupled to a thermal profile gener-ated by the optical heating. The thermal profile decaysat a much slower spatial rate and provides for long-rangeinteraction in what is effectively a three species, hyperbolic-parabolic, singularly perturbed system. We show thatby controlling the commutator of a scaling operator anda spectral projection, that the associated semi-group isstrongly contractive and use the RG machinary to deriveinteraction laws for N pulses comprised of optical pulse andlong-range thermal envelope.

Keith PromislowMichigan State [email protected]

MS40

A Simple Technique for Solving Partial DifferentialEquations on Surfaces


Steve RuuthSimon Fraser UniversityDepartment of [email protected]

MS40

HydroGel Swelling: Models and Analysis


Hang ZhangMichigan State [email protected]

MS41

Shock Reflection, Transonic Flow, and Free Bound-ary Problems

In this talk we will start with various shock reflection phe-nomena and their fundamental scientific issues. Then wewill describe how the shock reflection problems can be for-mulated into free boundary problems for nonlinear par-tial differential equations of mixed-composite type. Finallywe will discuss some recent developments in the study ofthe shock reflection problems. In particular, we will dis-cuss our recent results on the global theory for solutionsto shock reflection by wedges for potential flow. The ap-proach includes techniques to handle transonic shocks viafree boundary techniques, sonic curves via degenerate ellip-

PD07 Abstracts 69

tic techniques, and corner singularities when sonic curvesmeet transonic shocks that connect the subsonic phase withthe supersonic phase. Further remarks, trends, and openproblems in this direction will be also addressed. This talkis based mainly on our joint work with Mikhail Feldman.

Gui-Qiang G. ChenDepartment of Mathematics, Northwestern [email protected]

MS41

Symmetric Stationary Inviscid and Viscous Profilesin Multi-D

We construct stationary solutions with spherical or cylin-drical symmetry for the 2D and 3D compressible Euler andNavier-Stokes equations. The solutions are defined in theregion between two concentric spheres or cylinders and sat-isfy Dirichlet boundary conditions at the inner and outerboundaries. We first build stationary solutions with a sin-gle shock for the Euler equations (both isentropic and fullsystem). These are then used in the construction of cor-responding solutions of the Navier-Stokes equations. Inparticular, the viscous solutions converge to the inviscidshock solutions as the viscosities tend to zero.

Mark WilliamsDepartment of MathematicsUNC Chapel [email protected]

Kris JenssenPennsylvania State [email protected]

Erik EndresPennsylvania State UniversityDepartment of [email protected]

MS41

Shocks, Rarefactions and Triple Points in Multidi-mensional Conservation Laws

Following the initial work of Tesdall and Hunter [SIAP,2003] which found a new shock reflection pattern in theunsteady transonic small disturbance equations, we havenow exhibited this Guderley Mach reflection in numericalsimulations of a number of equations – specifically the non-linear wave system and the adiabatic gas dynamics equa-tions. Within this complicated pattern, the details of howthe rarefaction wave interacts with the sonic line form amathematically appealing subproblem. We present somenumerical and analytical results on this problem.

Barbara L. KeyfitzFields Institute and University of [email protected] or [email protected]

Allen TesdallSouthern Methodist [email protected]

Mary ChernFields Institutecher [email protected]

MS41

Instability Criteria in a Model for Shear Bands

We consider a system of hyperbolic-parabolic equations de-scribing the material instability mechanism associated tothe formation of shear bands at high strain-rates. We de-rive a quantitative criterion for the onset of instability:Using ideas from the theory of relaxation systems we de-rive equations that describe the effective behavior of thesystem. The effective equation turns out to be a forward-backward parabolic equation regularized by fourth orderterm.

Athanasios TzavarasUniversity of [email protected]

MS42

A Model-Based Inversion Algorithm for theControlled-Source Electromagnetic Data

We present the parametric inversion algorithm (PIA),which uses a priori information on the geometry to re-duce the number of unknown parameters and improve thequality of the reconstructed conductivity image. This PIAapproach can be also used to refine the conductivity im-age that we obtained using the pixel-based inversion (PBI)algorithm. The PIA adopts the Gauss-Newton minimiza-tion method, with nonlinear constraints and regularizationfor the unknown parameters. It also employs a line searchapproach to guarantee the reduction of the cost functionafter each iteration. The forward modeling simulation isa two-and-half dimensional finite-difference solver, and theparameters that govern the location and the shape of areservoir include the depth and the location of the user-defined nodes for the boundary of the region. The un-known parameter that describes the physical property ofthe region is the electrical conductivity. We will show somenumerical examples to illustrate the advantageous of usingthis PIA approach.

Tarek Habashy, Yan Zhang, Maokun LiSchlumberger-Doll [email protected], [email protected],[email protected]

Andrea [email protected]

Aria AbubakarSchlumberger Doll [email protected]

MS42



Aria AbubakarSchlumberger-Doll [email protected]

MS42

Optimal Reduced Models for Dispersive Multi-Conductor Transmission Line Systems

Through the use of non-uniform grids generated by Gaus-

70 PD07 Abstracts

sian spectral rules we develop reduced models for disper-sive, multi-conductor transmission line systems. The syn-thesized multi-port is represented as the concatenation ofsections of lumped circuits; hence, it is compatible withcircuit simulators such as SPICE. The passivity of the re-duced model is guaranteed through the use of passive cir-cuit representations of the frequency-dependent, per-unit-length series impedance and shunt admittance matrices ofthe transmission line system.

Aravind Ramachandran, Andreas CangellarisUniversity of Illinois at [email protected], [email protected]

MS42

Spectrally Matched Grids for Anistropic Problems

Spectral convergence of grids targeted for the Neumann toDirichlet map is observed for anisotropic problems, despitethe fact that the grids were designed for isotropic ones.We explain why this happens. For elliptic problems, thegridding algorithm is reduced to a Stieltjes rational ap-proximation on an interval of a line in the complex planeinstead of the real axis as in the isotropic case. We showrigorously why this occurs for a semi-infinite and boundedinterval. We then extend the gridding algorithm to hyper-bolic problems on bounded domains. For the propagativemodes, the problem is reduced to a rational approxima-tion on an interval of the negative real semiaxis, similarlyto in the isotropic case. For the wave problem we presentnumerical examples in 2-D anisotropic media.

Shari MoskowUniversity of FloridaDepartment of [email protected]

MS43

From Kinetic Fokker-Planck Traffic Models to Sec-ond Order Conservation Laws

We revisit multilane kinetic traffic models of Fokker-Plancktype and show that the Aw-Rascle second order model ofconservation type can be derived from such models. Fur-thermore, we present a linear stability analysis of the equi-libria associated with the (multivalued) fundamental dia-gram.

Reinhard IllnerThe University of Victoria, [email protected]

MS43

Analysis of a Fluid-Kinetic System of Equations

We study a coupled system of kinetic and fluid equa-tions modeling fluid-particles interactions arising in sprays,aerosols or sedimentation problems. More precisely, weconsider a Vlasov-Fokker-Planck equation coupled to com-pressible Navier-Stokes equation via a drag force. We es-tablish the existence of solutions and we rigorously derivethe asymptotic regime corresponding to a strong drag forceand a strong Brownian motion.

Antoine MelletUniversity of British [email protected]

MS43



Vladislav PanferovMcMaster UniversityDepartment of Mathematics and [email protected]

MS43

Recent Results on Transport in Random Media


Alexis VasseurUniversity of Texas at [email protected]

MS44

Existence and Stability of Ginzburg-Landau Vor-tices in Arbitrary Two Dimensional Domains

We consider Ginzburg-Landau equation with Neumann orDirichlet boundary conditions in a bounded domain. Weshow that whenever the renormalized energy W has criticalpoints (not necessary nondegenerate) then there exist so-lution to Ginzburg-Landau equation with vortices locatednear the set of critical points of W . We also show that theMorse index of these solutions is equal to the number ofthe negative eigenvalues of the hessian of the renormalizedenergy.

Manuel Del Pino, Michal KowalczykDepartamento de Ingeniera MatematicaUniversidad de [email protected], [email protected]

Monica MussoFacultad de MatematicasPontificia Universidad Catolica de [email protected]

MS44

A Gradient Flow Approach to a Mean Field Modelof Superconductivity

In a joint work with Luigi Ambrosio we studied anevolution problem proposed by Chapman-Rubinstein-Schatzman as a mean-field model for the evolution of thevortex density in a superconductor. We consider the case ofa bounded domain where vortices can exit or enter the do-main. We show that the equation can be derived rigorouslyas a gradient-flow of a specific energy for the Riemmanianstructure induced by the Wasserstein distance on probabil-ity measures. This leads to some existence and uniquenessresults and some energy-dissipation identities. We also ex-hibit some ”entropies” which decrease along the flow andallow to get regularity results.

Sylvia SerfatyCourant Institute, NYUNew York, [email protected]

MS45

Optimal Lower Bounds on the Stress Inside Ran-

PD07 Abstracts 71

dom Two-Phase Composites

A prescribed uniform stress is applied to the boundaryof a composite material made from two perfectly bondedisotropic linear elastic components. We present lowerbounds on the maximum stress generated inside the com-posite that apply to all mixtures made from two elasticmaterials in fixed volume fractions. We present several sit-uations in which these lower bounds are attained by simpleconfigurations of the two materials.

Bacim AlaliMathematics DepartmentLouisiana State [email protected]

Robert P. LiptonDepartment of MathematicsLouisiana State [email protected]

MS45

Gradient Estimates for the Perfect ConductivityProblem

We consider the optimal bound on the stress occurring be-tween a pair of closely spaced fibers in a bounded domain.With Dirichlet boundary condition, the conductivity prob-lem can be described as follows{

div{

[1 + χ(D1 ∪D2)(k − 1)]∇uk

}= 0 in Ω,

uk = ϕ on ∂Ω,

(1)where Ω contains two inclusions D1 and D2 which are ε-apart, and ∇uk represents the stresses.We study the per-fect conductivity problem, where k = +∞. We providean optimal blow-up rate of the gradient estimate as theseparation distance ε approaches 0. It is shown that theblow-up rate depends not only on the dimension, but alsoon the shape of the inclusions.

ShiTing Bao, YanYan Li, Biao YinMathematics DepartmentRutgers [email protected], [email protected],[email protected]

MS45

On Cohesive Fracture Without Relaxation

Cohesive models of fracture involve energies of the form

E(u) :=

∫Ω

W (∇u)dx +

∫K

φ([u])dHn−1,

where u is a deformation with crack set (i.e., discontinuityset) K and [u] is the jump in u across K. What makes themodel cohesive is that φ(x) → 0 as x → 0, so that thereare forces across K if [u] is small enough. If φ is concavewith φ′(0) = ∞, then existence of a minimizer is straight-forward. However, if φ′(0) < ∞, there at first seems tobe a fatal issue in that one needs to relax the energy tofunctions u that have singular behavior besides disconti-nuities (i.e., relaxed from SBV to BV ), resulting in theaddition of a new term to the energy. Precisely, a sequence{ui} in SBV with bounded energy might converge to afunction u /∈ SBV , which appears fatal to applying thedirect method of the calculus of variations. However, thereis reason to believe that minimizing sequences cannot have

this property, and in fact, minimizers of the relaxed energymust be in SBV . I will describe work in this direction,which is joint with G. Dal Maso and A. Garroni.

Christopher LarsenWorcester Polytechnic InstituteMath [email protected]

MS45

On the Eshelby Conjecture and a Symmetry Resultfor Overdetermined Elliptic Problems

In this paper we present solutions to the Eshelby conjec-ture. We first discuss the exact meanings of the originalEshelby conjecture. We then prove one version of the Es-helby conjectures and construct a counterexample to theother by exploring a related variational inequality problem.In particular we construct multiply-connected regions in alldimensions and nonellipsoidal connected regions in threeand higher dimensions that share the special property ofan ellipsoid that some uniform eigenstress of the region in-duces uninform strain on the region. We also show that oneversion of the Eshelby conjecture is equivalent to assertingthat some overdetermined elliptic problem has a solution ifand only if the region is an ellipsoid. We therefore obtaina symmetry result for this overdetermined elliptic problemby studying the Eshelby conjecture.

Liping LiuMechanical Engineering DepartmentCalifornia Institute of [email protected]

MS46

Stable Numerical Schemes for Constitutive Equa-tions of Complex Fluids

Modelling of polymeric fluids can be done using two ap-proaches: macro-macro formulations or micro-macro for-mulations using kinetic equations to model the evolutionof the microstructures in the fluid. Macro-macro modelsare cheaper to discretize than micro-macro models, butare known to blow up when a parameter, the Weissenbergnumber, increases. Many studies (e.g. [1]) have identifiedpossible reasons for that so-called High Weissenberg Num-ber Problem (HWNP), but have not led yet to a completeunderstanding of the numerical instabilities. Micro-macromodels often yield more information and better under-standing of the coupling between scales than macro-macromodels. The key role of the Free energy in long-time be-haviour was first understood thanks to the equivalence ofthe Oldroyd-B model with a kinetic model [2]. We willdiscuss the stability of finite element methods to discretizemacro-macro models, using Free energy estimates. [1] R.Fattal and R. Kupferman, Time-dependent simulation ofviscoelastic flows at high Weissenberg number using thelog-conformation representation, J. Non- Newtonian FluidMechanics, 126 (1) (2005) 23-37. [2] B. Jourdain and C. LeBris and T. Lelievre and F. Otto, Long-Time Asymptoticsof a Multiscale Model for Polymeric Fluid Flows , Archivefor Rational Mechanics and Analysis, 181 (2006) 97-148.

Sebastien BoyavalCERMICSEcole Nationale des Ponts et [email protected]

Tony Lelievre

72 PD07 Abstracts

[email protected]

MS46

Global Orientation Dynamics for Liquid Crys-talline Polymers

We study global orientation dynamics of the Doi-Smoluchowski equation with the Maier-Saupe potential onthe sphere, which arises in the modeling of rigid rod-likemolecules of polymers. Using the orientation tensor wefirst reconfirm the structure and number of equilibrium so-lutions established in [Comm. Math. Sci. 3 (2), (2005),201-218] by H. Liu, H. Zhang and P.-W. Zhang. We thenexamine global orientation dynamics in terms of eigenval-ues of the orientation tensor via the Doi closure approxi-mation. It is shown that for small intensity 0 < α < 4, allstates will evolve into the isotropic phase; for large inten-sity α > 4.5, all states will evolve into the nematic prolatephase; and for the intermediate intensity 4 < α < 4.5, aninitial state will evolve into either the isotropic phase orthe nematic prolate phase, depending on whether such aninitial configuration crosses a critical threshold. Moreover,the uniaxial symmetry structure is shown to be preservedin time.

Hailiang LiuIowa State [email protected]

MS46

On the New Multiscale Rod-Like Model of Poly-meric Fluids

It is concerned with the well-posedness for the new rigidrod-like model in a polymeric fluid and the structure ofsome solutions in special case. The constitutive relationsconsidered in this work are motivated by the kinetic theory.The micro equation has five spatial freedom variables, twoof them are in the configuration domain and the others arein the macro flow domain.

Hui ZhangBeijing Normal [email protected]

MS46

Dynamical Modeling and Simulation of Fluid Bio-Membranes

This work provides a set of equations for the dynamics ofevolving fluid membranes, such as cell membranes, in thepresence of bulk fluids. We model the membrane as a sur-face endowed with a director field, which describes the localaverage orientation of the molecules on the membrane. Amodel for the elastic energy of a surface endowed with adirector field is derived using liquid crystal theory. Thiselastic energy reduces to the well-known Helfrich energyin the limit when the directors are constrained to be nor-mal to the surface. We then derive the full dynamic equa-tions for the membrane that incorporate both the elasticand viscous effects, with and without the presence of thebulk fluids. We also consider the effect of local sponta-neous curvature, arising from the presence of membraneproteins. Overall the new systems of equations allow usto carry out stable, accurate and robust numerical model-ing for the dynamics of the membranes. In addition, we

consider the coupling between the bulk fluid and the mem-brane dynamics. Dynamic structure factors are computedin varies cases. These new systems of equations allow us tocarry out stable and robust numerical simulations for thedynamics of the membranes.

Pingwen ZhangPeking [email protected]

MS47

Asymptotics of the Semiclassical Sine-GordonEquation

The study of the sine-Gordon equation with a fixed ini-tial condition and dispersion tending to zero is motivatedby the modeling of magnetic flux propagation in Josephsonjunctions. We discuss the recent discovery of two families ofinitial data, with topological charge zero and one, respec-tively, for which the sine-Gordon equation can be solvedexplicitly for arbitrarily small dispersion. Plots of the solu-tions for small dispersion reveal, depending on the choice ofother parameters, regions of pure librational and rotationalmotion, as well as regions of multi-phase waves separatedby primary and secondary nonlinear caustics. We presentcurrent progress on the asymptotic analysis of these solu-tions for small dispersion.

Robert J. BuckinghamUniversity of [email protected]

MS47

Universality Classes for Semiclassical EigenvalueProblems

We will discuss some generalized eigenvalue problems (inwhich the eigenvalue does not necessarily enter linearly) ina semiclassical scaling (where derivatives are multiplied bysmall coefficients). Such problems arise frequently in theasymptotic analysis of nonlinear problems solvable by aninverse-scattering transform. We will show how the asymp-totics of the discrete spectrum leads to the idea of univer-sality classes of potentials and describe how this idea canbe used to approximate the discrete spectrum with quan-titative error estimates.

Peter D. MillerUniversity of [email protected]

MS47

Semiclassical Limit Scattering Map (Direct and In-verse ) for the Focusing Nonlinear SchroedingerEquation.

We discuss the progress in finding the semiclassical (zerodispersion) limit of the scattering map for the focusingNLS, that put into correspondence the initial potentialq(x, ε) and the scattering data. In the solitonless case, thescattering data consists of the reflection coefficient r(z, ε),where z is the spectral and ε is the semiclassical parameterrespectively. So, in this case, we are considering the ε → 0limit of the maps q �→ r and r �→ q.

Stephanos VenakidesDuke UniversityDepartment of [email protected]

PD07 Abstracts 73

Alexander TovbisUniversity of Central FloridaDepartment of [email protected]

Xin ZhouDuke [email protected]

MS47

Persistence of Modulated Post-Break NLS Wave-forms in the Presence of Solitons.

We analyze the persistence of modulated 2-phase NLSwaves developed after the first break (first nonlinear caus-tic line in space-time). In previous work, we have shown(within an initial wave structure) global time-persistence ofthese waves when the initial data are solitonless. We nowshow that, for initial data containing solitons, the same istrue for all space points whose distance from the origin isless than a calculated distance d. For spatial points closerto the origin, we calculate an asymptotic formula (as |x|increases to d) for time t = t(x) up to which the waveformis guaranteed not to undergo a second break.


Alexander TovbisUniversity of Central FloridaDepartment of [email protected]

Xin Zhou, Sergey BelovDuke [email protected], [email protected]

MS48

Wiener’s Criterion for the Regularity of the Pointat Infinity and its Measure-Theoretical Counter-part

We introduce a notion of regularity (or irregularity) of thepoint at infinity for the unbounded open set concerning theheat equation, according as whether the parabolic measureof the point at infinity is zero or positive. We prove thenecessary and sufficient condition for the existence of aunique bounded solution to the parabolic Dirichlet problemin an arbitrary unbounded open set expressed in terms ofthe Wiener’s criterion for the regularity of the point atinfinity.

Ugur G. AbdullaDepartment of Mathematical SciencesFlorida Institute of [email protected]

MS48

Connecting Non–Negative Solutions to Quasi–Linear Degenerate Equations, Sub-potentials

Non–negative weak solutions of quasilinear degenerateparabolic equations of p–Laplacian type are shown to be lo-cally bounded below by sub–potentials. As a consequencenon–negative solutions expand their positivity set and sat-

isfy alternative, equivalent forms of the Harnack inequality

Emmanuele DiBenedettoDepartment of MathematicsVanderbilt [email protected]

MS48

Continuity of the Saturation in the Flow of TwoImmiscible Fluids Through a Porous Medium

We consider a weakly coupled, highly degenerate systemof an elliptic equation and a parabolic equation, arising inthe theory of flow of immiscible fluids in a porous medium.The unknown functions in the system are u and v, andrepresent pressure and saturation respectively. By usingsuitable DeGiorgi’s techniques, we prove that the satura-tion is a continuous function in the space–time domain ofdefinition. Here the main technical point is that no as-sumptions at all are made on the nature of the degeneracyof the system.

Vincenzo VespriDepartment of MathematicsUniversity of [email protected]

Ugo P. GianazzaDepartment of MathematicsUniversity of [email protected]

Emmanuele DiBenedettoDepartment of MathematicsVanderbilt University, [email protected]

MS48

Beyond the Laplacian

In this talk we discuss some applications of boundary Har-nack inequalities for positive p-harmonic functions (re-cently proved by Lewis and Nystrom) vanishing on theboundary of a Lipschitz or Reifenberg flat domain. Pos-sible topics include the dimension of p-harmonic measurefor Wolff snowflake, the construction of p-harmonic qua-sispheres, and codimension ¿ 1 Harnack - free boundaryregularity.

John LewisUniversity of [email protected]

MS48

On the Regularity of the Free Boundary for theClassical Stefan Problem

The regularity of the temperature and the free boundaryfor the Stefan problem has attracted much attention overthe last 3 decades or so. In this talk I will consider theclassical one- or two-phase Stefan problem, where the freeboundary is represented as the graph of a function. Un-der mild regularity assumptions on the initial data it willbe shown that the problem admits a local solution thatis analytic in space and time. The approach is based onoptimal regularity results for a linearized problem, and ona scaling-translation argument in conjunction with the im-

74 PD07 Abstracts

plicit function theorem.

Gieri SimonettVanderbilt [email protected]

Jan PrussMartin-Luther University [email protected]

Jurgen SaalUniversity of Konstanz, [email protected]

MS49

On Long-TimeDynamics for Competition-Diffusion Systems withInhomogeneous Dirichlet Boundary Conditions

We consider a two-component competition-diffusion sys-tem with equal diffusion coefficients and inhomogeneousDirichlet boundary conditions. When the interspecificcompetition parameter tends to infinity, the system solu-tion converges to that of a freeboundary problem. If all sta-tionary solutions of this limit problem are non-degenerateand if a certain linear combination of the boundary datadoes not identically vanish, then for sufficiently large in-terspecific competition, all non-negative solutions of thecompetition-diffusion system converge to stationary statesas time tends to infinity. Such dynamics are much simplerthan those found for the corresponding system with eitherhomogeneous Neumann or homogeneous Dirichlet bound-ary conditions.

Norman DancerSchool of Mathematics and StatisticsUniversity of [email protected]

Danielle HilhorstLaboratoire de MathematiquesUniversite de [email protected]

Elaine CrooksOxford [email protected]

MS49

High Lewis Number Combustion Fronts: a Geo-metric Singular Perturbation Theory Point of View

I will discuss combustion wavefronts that arise in a math-ematical model for high Lewis number combustion pro-cesses. An efficient method for the proof of the existenceand uniqueness of combustion fronts is provided by geo-metric singular perturbation theory. The fronts supportedby the model with very large Lewis numbers are small per-turbations of the front supported by the model with infiniteLewis number. The question of stability of fronts is morecomplicated. I will describe the issues arising in the sta-bility analysis and up to date known results, in particular,how a geometric approach which involves construction ofthe augmented unstable bundles can be used to relate thespectral stability of the wavefront with high Lewis num-ber to the spectral stability in the case of infinite Lewisnumber. This is a joint work with C.K.R.T. Jones

Anna Ghazaryan

The University of North Carolina at Chapel [email protected]

MS49

Propagation of a KPP-Type Fronts in a Model ofPressure Driven Flames

We study the model of pressure driven-flames in porousmedia introduced by G. Sivashinsky et. al. Under theassumption that the chemical kinetics is described by theKPP-type nonlinearity, we show that the problem admits afamily of positive traveling solutions. Moreover, it is shownthat the long time behavior of the model is prescribed bythe rate of decay of initial data at infinity. This is a jointwork with A. Ghazaryan.

Peter GordonNew Jersey Institute of [email protected]

MS49

Blow Up and Regularity for Burgers Equation withFractional Dissipation

I will discuss recent results on regularity and blow up ofsolutions of Burgers equation with fractional dissipation.Fluid mechanics models with fractional dissipation, such assurface quasi-geostrophic equation, have recently attractedsignificant attention. The Burgers equation is perhaps thesimplest model for studying interaction between nonlinear-ity and fractional viscosity. We establish a sharp transitionbetween singular front formation and smooth evolution asthe fractional dissipation exponent passes 1/2. We alsoprove analyticity of the solutions in the subcritical case,and prove existence of solutions for rough initial data.

Alexander KiselevUniversity of Wisconsin, [email protected]

MS50

Global Well-Posedness of the Three-DimensionalGeostrophic Turbulence Mixing Model

The three-dimensional viscous Boussinesq equations underhydrostatic balance govern large scale dynamics of atmo-sphere and oceanic motion. To overcame the turbulencemixing a vertical diffusion is added. In this paper we provethe global existence and uniqueness (regularity) of strongsolutions to this model.

Edriss TitiDepartment of MathematicsUniversity of California, [email protected]

Chongsheng CaoDepartment of MathematicsFlorida International [email protected]

MS50

Wave Turbulence: A Story Far from Over

Wave turbulence, the study of the long time statisticalbehavior of solutions of nonlinear field equations describ-ing weakly nonlinear dispersive waves in the presence ofsources and sinks, is a subject far from over. Among the

PD07 Abstracts 75

remaining challenges are: 1. Since wave turbulence theoryalmost never holds at all scales, one has to understand howthe wave turbulence stationary states coexist with fullynonlinear states. 2. Understanding how the Kolmogorov-Zakharov spectra are realized in cases where these solutionshave finite capacity. The results of 2. may very well haverelevance for turbulence in general. I will briefly discussthese challenges and suggest some resolutions.

Alan NewellUniversity of ArizonaDepartment of [email protected]

MS50

Daynimic Transitions in Geophysical Fluid Dynam-ics

First we shall present a new dynamic transition theory fornonlinear dynamical systems, both finite and infinite di-mensional. The theory will be used to 1) classify dynamictransitions of different flow regimes of rotating Boussinesqequations, a basic model in geophysical fluid dynamics, and2) to provide a new theory for ENSO.

Tian MaSichuan [email protected]

Shouhong WangDepartment of MathematicsIndiana [email protected]

MS50

The Surface Quasi-Geostrophic Equation

This talk reports several recent results concerning solutionsof the inviscid quasi-geostrophic (QG) equation and of thedissipative QG equation with supercritical dissipation.

Jiahong WuOklahoma State [email protected]

MS51

Diffuse Interface Methods for Image Reconstruc-tion

We consider some recent diffuse interface methods for im-age reconstruction. Applications include image inpainting,in which missing information must be recovered in some re-gion, and deconvolution, in which an image is blurred dueto the process of obtaining the data. Real life applicationsinclude document explotation, following roads in satteliteimagery, and bar code scanners. We present some newmethods with fast solvers based on diffuse interface equa-tions from materials science. One class of methods involvesthe Cahn-Hilliard equation for binary mixtures. Anotherclass of methods involves a nonlinear wavelet-based func-tional related by scaling to the Ginzburg-Landau energyfrom diffuse interface methods. We consider both binaryand greyscale applications with possible extension to highdimensional pixel data. This is joint work with Julia Do-brosotskaya, Selim Esedoglu, and Alan Gillette.

Andrea L. BertozziUCLA Department of [email protected]

MS51

Some Numerical Results on Rudin-Osher-FatemiSmoothing

We present some numerical algorithms, their analyses, andresults for ROF-style variational image smoothing.

Bradley LucierDepartment of MathematicsPurdue [email protected]

MS51

Some Properties of a Perturbed Version of thePerona-Malik Equation

The one-dimensional version of Perona-Malik equation,originally proposed for image selective smoothing, is con-sidered. This is an ill-posed parabolic equation for whichstable numerical schemes exist. We investigate a perturbedversion of the equation such that the perturbation doesnot change the order of the differential operator. The per-turbed problem is well posed and the asymptotic behaviourof its solutions is analyzed when a suitable perturbationparameter tends to zero.

Riccardo MarchIstituto per le Applicazioni del Calcolo, [email protected]

Mario Rosati, Andrea SchiaffinoUniversita’ Roma [email protected], [email protected]

MS51

Curvature of Level Sets of BV Functions


Murali RaoUniversity of [email protected]

MS51

(Phi,Phi*) Image Decomposition Models and Min-imization Algorithms

We review minimization algorithms for image restorationand decomposition using dual functionals and dual norms.In order to extract a clean image u from a degraded versionf=Ku+n, where f is the observation, K is a blurring opera-tor and n represents additive noise, we impose a standardregularization penalty on u, of the form Phi(u), dependingon the gradient of u; however, on the residual f-Ku, we im-pose a weaker dual penalty Phi*(f-Ku), instead of the morestandard L2 data fidelity term. In particular, when Phi(u)is the total variation of u, we recover the (BV,BV*) decom-position of the data f, as suggested by Y. Meyer. Practicalminimization methods are presented, together with exper-imental results and comparisons to illustrate the validityof the proposed models.

Luminita A. VeseUniversity of California, Los AngelesDepartment of [email protected]

Triet M. Le

76 PD07 Abstracts

Yale [email protected]

Linh M. LieuUniversity of California, [email protected]

MS52

A Variational Model for Bent-Core Liquid Crystals

We present a nonlinear variational model for bent-core(banana-shaped) liquid crystals which contains energyterms including polarization, smectic aned chiral effects,elasticity, and surface tension. We analyze solutions in aphysically realistic regime involving a free boundary prob-lem for the energy and the existence of a Gamma limit.

Daniel PhillipsDepartment of Mathematics, Purdue [email protected]

Patricia BaumanPurdue [email protected]

MS52

Equilibrium Configurations of Epitaxially StrainedCrystalline Films: Existence and Regularity Re-sults

Strained epitaxial films grown on a relatively thick sub-strate are considered in the context of plane linear elastic-ity. The total free energy of the system is assumed to bethe sum of the energy of the free surface of the film andthe strain energy. Due to the lattice mismatch between filmand substrate, flat configurations are, in general, energeti-cally unfavorable and a corrugated or islanded morphologyis the preferred growth mode of the strained film. New reg-ularity results for volume constrained local minimizers ofthe total free energy are established, leading to a rigorousproof of the zero contact-angle condition between islandsand wetting layers.

Irene FonsecaCarnegie Mellon UniversityCenter for Nonlinear [email protected]

MS52

A Gamma-Convergence Approach to the Cahn-Hilliard Equation

We study the asymptotic dynamics of the Cahn-Hilliardequation via the “Gamma convergence” of gradient flowsscheme initiated by Sandier and Serfaty. This gives riseto an H1-version of a conjecture by De Giorgi, namely,the slope of the Allen-Cahn functional with respect tothe H−1-structure Gamma-converges to a homogeneousSobolev norm of the scalar mean curvature of the limit-ing interface. We confirm this conjecture in the case ofconstant multiplicity of the limiting interface. Finally, un-der suitable conditions for which the conjecture is true,we prove that the limiting dynamics for the Cahn-Hilliardequation is motion by Mullins-Sekerka law.

Nam LeCourant Institute, [email protected]

MS52

Lorentz Space Estimates for Ginzburg-Landau Vor-tices

We discuss the Ginzburg-Landau model of superconductiv-ity in two dimensions. We present a technical improvementof the “vortex balls construction” that allows for the ex-traction of a new positive term in the energy lower boundsin the vortex balls. This term is then estimated using theLorentz space L2,∞, which is critical for the expected vor-tex profiles. From this we can estimate the total numberof vortices and prove improved convergence results.

Ian TiceCourant Institute, [email protected]

MS53

Monge’s Mass Transportation in the HeisenbergGroup

Abstract not available at time of publication

Robert BerryUniversity of [email protected]

MS53



Nicola GarofaloPurdue [email protected]

MS53

Regularity of Lipschitz Free Boundaries in Two-Phase Problems for the p-Laplace Operator

In this paper we study the regularity of the free bound-ary in a general two-phase free boundary problem for thep-Laplace operator and we prove, in particular, that Lips-chitz free boundaries are C1,γ-smooth for some γ ∈ (0, 1).As part of our argument, and which is of independent in-terest, we establish the Hopf boundary principle for non-negative p-harmonic functions vanishing on a portion ofthe boundary of a Lipschitz domain.

Kaj NystromUmea University, [email protected]


MS53

P-Harmonic Measure on Wolff Snowflakes.

We study the dimension of p-harmonic measure on Wolffsnowflakes using some of the new results on boundary Har-nack inequalities of Lewis and Nystrom.

Kaj NystromUmea University, [email protected]

PD07 Abstracts 77


Andrew VogelSyracuse [email protected]

Bjorn BennewitzUniversity of [email protected]

MS54

Rheology of Bacterials Suspensions

Modeling of bacterial suspensions and, more generally, ofsuspensions of active microparticles has recently become anincreasingly active area of research. The focus of this workis on the development and analysis a mathematical PDEmodel for a multiscale problem of bacterial suspensions.We start from a comparison of bacterial suspensions (activesuspensions) and suspensions of solid particles (passive sus-pensions). The ultimate goal is to develop a model whichdescribes large scale pattern formationand rhelogical prop-erties of bacterial suspensions. We discuss recent resultson effective viscosity of dulite bacterial suspensions (withAronson, Haines and Karpeev) and asymptotic analysis ofswimming patterns of bacteria (with Aronson, Gyrya andKarpeev).


MS54

New Exact Bounds for Effective Conductivity ofMultiphase Composites and Optimal Microstruc-tures


Andrej V. CherkaevDepartment of MathematicsUniversity of [email protected]

MS54

Homogenization and Field Concentrations in Het-erogeneous Media

A method for upscaling the local field concentrations insiderandom composite and polycrystalline media is presented.The talk focuses on gradient or strain fields associated withsolutions of second order elliptic PDE used in the descrip-tion of thermal transport and elasticity inside random me-dia. We develop a method for assessing the Lp integrabilityof gradient and strain fields inside microstructured media.The results are described in terms of the pth order momentsof the solution of two-scale corrector problems. Examplesare provided that illustrate the theory and its application.

Robert P. LiptonDepartment of MathematicsLouisiana State [email protected]

MS54

Darcy-Brinkman Models of Fast Channel Flow atan Interface

We show that preferential flow or channeling effects at theboundary of a porous medium can be effectively modeledby a Darcy-Brinkman interface system.

Fernando MoralesOregon State [email protected]

Ralph ShowalterDepartment of MathematicsOregon State [email protected]

MS55

Mathematical Problems Arising in Near-ShoreZone Sediment Trasport

A wave - sediment interaction model is put forward andvalidated. Some mathematical issues that arise as a conse-quence will be described.


MS55

Water Waves Over Bottom Topography

We present a Hamiltonian formulation for water waves overbottom topography based on potential flow theory. In thisformulation, the problem is reduced to a lower-dimensionalsystem involving boundary variables alone. This is ac-complished by introducing the Dirichlet-Neumann opera-tor which expresses the normal fluid velocity at the freesurface in terms of the velocity potential there, and interms of the surface and bottom variations which deter-mine the fluid domain. A Taylor series expansion of theDirichlet-Neumann operator in powers of the surface andbottom variations is proposed. This formulation has im-plications for the convenience of perturbation calculationsand numerical simulations of the full Euler equations. Wedevelop an efficient and accurate spectral method basedon the fast Fourier transform to evaluate numerically theDirichlet-Neumann operator and solve the full Euler equa-tions. Tests and validation will be shown. This is jointwork with W. Craig (McMaster University), D. P. Nicholls(UIC) and C. Sulem (University of Toronto).

Philippe GuyenneUniversity of [email protected]

MS55

Long Nonlinear Internal Waves Models

We derive systematically and justify rigorously asymptoticmodels for the propagation of long nonlinear internal wavesin 2+1 dimensions. This is a joint work with Jerry Bonaand David Lannes.

Jean-Claude [email protected]

78 PD07 Abstracts

MS56

On a Variational Approach for Stokes Conjectureon Water Waves

We study the behaviour of a two-dimensional, inviscid, ir-rotational fluid in a domain of finite depth and in the pres-ence of gravity. More precisely, we investigate the mini-mizers of the functional

Jλ (u) =

∫Ω

(|∇u|2 + (λ− y)+ χ{u>0}

)dx dy

with Dirichlet boundary condition at the bottom. We showthat for large values of the paremeter λ, minimizers of Jλ

have a regular (analytic) free boundary while for small val-ues of λ, minimizers are ”nonphysical”. This leads to acritical value of λ related to Stokes wave of greatest height.

Danut AramaCarnegie Mellon UniversityPittsburgh, PA [email protected]

MS56

A Variational Formulation for Level Set Represen-tation of Multiphase Flow.

We discuss variational formulations for level set represen-tations of various multiphase flows that involve curvature.Our formulations lead to practical algorithms for comput-ing these flows accurately. Joint work with Peter Smereka.

Selim EsedogluDepartment of MathematicsUniversity of [email protected]

MS56

Phase Transitions in Thin Films

In this talk, I will present some models for solid-solid phasetransitions in thin films obtained by Γ-convergence. Thedifferent models are determined by the asymptotic ratio be-tween the characteristic length scale of the phase transitionand the thickness of the film. Depending on the regime, Iwill show that a separation of scales holds or not and howthis later case is related to some rigidity results. This is ajoint work with V. Millot.

Bernardo Galvao-SousaCarnegie Mellon [email protected]

MS56

Optimal Transport for the System of Isentropic Eu-ler Equations

In this talk we report on a variational time discretizationfor the system of isentropic Euler equations. The schemeis inspired by the theory of abstract gradient flows on thespace of probability measures, and by Dafermos’ entropyrate criterion: The total energy should be decreased atmaximal rate. This is joint work with Wilfrid Gangbo.

Michael WestdickenbergGeorgia Institute of [email protected]

MS56

Pulsating Wave for Motion by Mean Curvature inInhomogeneous Medium

We prove the existence and uniqueness of pulsating wavesfor the motion by mean curvature of a hypersurface in aninhomogeneous medium given by periodic forcing. Themain difficulty is caused by the degeneracy of the equa-tion and the fact the inhomgeneity is allowed to change itssign. Under the assumption of weak forcing, we obtain uni-form oscillation and gradient bounds so that the evolvingsurface can be represented as a graph over a hyperplane.The existence of an effective speed of propagation is es-tablished for any normal direction. We further prove theLipshitz continuity of the speed with respect to the nor-mal and various stability properties of the pulsating wave.Some connection with homogenization theory will also bediscussed. This is joint work with N. Dirr and G. Karali.

Aaron YipPurdue UniversityDepartment of [email protected]

MS57

Ginzburg-Landau Vortices Concentrating onCurves

We study a two-dimensional Ginzburg-Landau functional,which describes superconductors in an externally appliedmagnetic field. We are interested in describing the en-ergy minimizers at the critical value of the magnetic fieldfor which vortices first appear in the superconductor (the”lower critical field”.) The vortices are quantized singu-larities, and we are interested in their number and theirdistribution in the sample nearby the critical field. I willdescribe recent results with L. Bronsard and V. Millot inwhich we study the number and distribution of these vor-tices which concentrate along a curve. Their distributionis determined by a classical problem from potential theory.

Stanley AlamaMcMaster UniversityDepartment of Mathematics and [email protected]

Vincent MillotCarnegie Mellon UniversityPittsburgh, PA [email protected]

Lia BronsardMcMaster [email protected]

MS57

Global Minimizers of the LawrenceDoniach Modelwith Oblique External Field

We study periodic minimizers of the Lawrence-Doniachmodel for layered superconductors, in various limitingregimes. We are particularly interested in determining thedirection of the internal magnetic field (and vortex lattice)as a function of the applied external magnetic strength andits orientation with respect to the superconducting planes.We identify the corresponding lower critical fields, andcompare the Lawrence-Doniach and anisotropic Ginzburg-Landau minimizers in the periodic setting. This talk rep-

PD07 Abstracts 79

resents joint work with S. Alama and E. Sandier.

Stanley AlamaMcMaster UniversityDepartment of Mathematics and [email protected]

Etienne SandierUniv. Paris 12Creteil, [email protected]

Lia BronsardMcMaster UniversityDepartment of [email protected]

MS57

Analysis of Heterogeneous Superconducting Sys-tems Via the Ginzburg-Landau Approach.

Generalizations of the Ginzburg-Landau energy functionalhave been introduced in the physics literature to describecomposite superconducting bodies. For example, the pres-ence of a metal or semiconductor that surrounds or coats asuperconductor can be modeled by a suitable extension ofthe energy to the non-superconducting components or bythe addition of surface integrals. We will present analyticalresults which illustrate how effective these models can bein describing the physical response of the systems underexamination.

Tiziana GiorgiNew Mexico State UniveristyDepartment of Mathematicstgiorgi@nmsu

MS57

Vortex Lines in Bose-Einstein Condensates

In this talk I will present some results that give a prelimi-nary description of the vortex lines exhibited by Bose Ein-stein condensates, as well as super-conductors, subject toa stirring in an appropriate range. What one finds is thatthese vortex lines can be described by means of an ODEthat represents the equation of the geodesics in an appro-priate manifold. This is joint work with Ben Stephens.

Alberto MonteroUniversity of TorontoDepartment of [email protected]

MS58

Strong Convergence to Homogeneous CoolingStates

We study solutions of the spatially homogeneus Boltzmannequation with inelastic collisions with rescaling to fix thetemperatures. We prove regularity properties for these so-lutions that are valid uniformly in time, and use thses toquantify the rate of convergence to the homogeneous cool-ing state in the strong L1 norm. This is joint work withM.C. Carvalho and J. Carrillo.

Eric A. CarlenGeorgia [email protected]

MS58

Propagation of Fisher Information for InelasticCollisions

We prove bounds on the Fisher information for solutions ofthe spatially homogeneus Boltzmann equation with inelas-tic collisions. The bounds diverge as time tends to infinity,as they must, due to the dirac mass limit, but nonetheless,the sharp bound obtained here plays a key role in a proofthat when the solutions are rescaled to preserve the secondmoment, the result is regular uniformly in time. This isjoint work with E. Carlen and J. Carillo.

Maria CarvalhoUniversity of Lisbon and Georgia [email protected]

MS58

Kinetic Dynamics of Multilinear Stochastic Inter-actions


Irene M. GambaDepartment of Mathematics and ICESUniversity of [email protected]

MS58

Spectral - Lagrangian Based Deterministic Methodfor Space Inhomogenous Boltzmann Equation.

A new spectral Lagrangian based deterministic solver forthe non-linear Boltzmann Transport Equation for VariableHard Potential VHP) modeling collision kernels has beenproposed for both conservative and non-conservative bi-nary interactions. The method is based on the symmetriesof the Fourier transform of the collision integral, where thecomplexity in computing it is reduced to a separate inte-gral over the unit sphere S2. In addition the conservationof moments is enforced by Lagragian constrains. The re-sulting scheme is very versatile and adjust to in a verysimple manner to several that involved energy dissipationdue to local micro-reversibility (inelastic interactions) orelastic model of slowing down process. The issue of con-vergence has also been addressed. Since the equation issolved for explicit distribution function, the high velocitytail behavior can be analyzed explicitly. Such an analysisis crucial in understanding the physical processes involvedin certain problems such as boundary and strong shocklayer formation and sheared flows in classical elastic di-luted gases or rapid granular flows or a mixture of gaseswhere classical fluid dynamical models fail.The proposednumerical method has been used to solve the space inho-mogenous Boltzmann equation for Riemann problem andCouette flow problems.

Sri Harsha TharkabhushanamThe University of Texas at [email protected]

MS59

Some Mathematical Models in Emerging Biomedi-cal Imaging


Habib AmmariCNRS & Ecole Polytechnique, France

80 PD07 Abstracts

[email protected]

MS59

Modeling and Design of Coated Stents in Interven-tional Cardiology

Stents are used in interventional cardiology to keep a dis-eased vessel open. New stents are coated with a medicinalagent to prevent early reclosure due to the proliferation ofsmooth muscle cells. It is recognized that it is the dose ofthe agent that effectively controls the growth. This paperfocusses on the asymptotic behavior of the dose for gen-eral families of coated stents under a fixed ratio betweenthe coated area of the stent and the targeted area of thevessel and set therapeutic bounds on the dose. It gener-alizes the results of Delfour, Garon, and Longo (SIAM J.Appl Math) for stents made of a sequence of thin equallyspaced rings to stents with an arbitrary pattern. It givesthe equation of the asymptotic dose for a normal tiling ofthe target region.

Michel C. DelfourCentre de recherches mathematiquesUniversite de Montreal, [email protected]

MS59

Optimization and Control Techniques for MedicalImage Registration


Gunter LeugeringUniversity Erlangen-NurembergInstitute of Applied [email protected]

MS59

Drag Minimization for Stationary, CompressibleNavier-Stokes Equations


Jan SokolowskiInstitut Elie CartanUniversite Henri Poincare Nancy [email protected]

Pavel PlotnikovLavryentyev Institute of HydrodynamicsNovosibirsk, [email protected]

MS60

Defect Eigenvalues and Diophantine ConditionsVia the Evans Function.

We consider Schrodinger eigenvalue problem in one dimen-sion with a potential consisting of a compactly supported”defect” potential plus a periodic part. This is a model forsimilar structures in optics. We give a result which allowsone to count the number of point eigenvalues which are cre-ated in a gap in the periodic spectrum. This count can beexpressed either in terms of an associated purely periodicspectral problem or as the Maslov index of a certain curvein the symplectic group. This count is always within one ofthe actual number of eigenvalues (in either direction) and

is frequently exact.

Jared BronskiUniversity of Illinois Urbana-ChampainDepartment of [email protected]

MS60

Dispersion Relations for Subwavelength PhotonicCrystals

We examine wave propagation in sub-wavelength dielec-tric/metal photonic crystals near the high plasma fre-quency limit. We characterize the band structure for thiscase. Examples are presented for layered media and waveguides.

Stephen P. Shipman, Robert P. Lipton, Santiago FortesDepartment of MathematicsLouisiana State [email protected], [email protected],[email protected]

MS60

Fano Resonance in Photonic Crystal Slabs

Spectrally embedded guided modes in photonic crystalslabs are nonrobust under perturbations of the structureor the wave vector. When the eigenvalue for the nonrobustmode dissolves into the continuous spectrum, it leaves aspectral region of anomalous transmission in its wake. Thisphenomenon has its analog in the Fano resonances in quan-tum mechanics. Our rigorous asymptotic formula for thetransmission anomalies for periodic slabs is a generalizationof the Fano formula for line shapes in physical chemistry.Joint work with S. Venakides.

Stephen P. ShipmanDepartment of MathematicsLouisiana State [email protected]


MS61

Schroedinger Maps (Landau-Lifschitz equation)


Stephen GustafsonDepartment of MathematicsUniversity of British [email protected]

MS61

On the Lifetime of Quasi-Stationary States in Non-Relativistic QED

We consider resonances in the Pauli-Fierz model of non-relativistic QED. We use and slightly modify the analysisdeveloped by V. Bach, J. Froehlich, and I.M. Sigal to ob-tain an upper and lower bound on the lifetime of quasi-stationary states. This is joint work with I. Herbst and M.

PD07 Abstracts 81

Huber.

David HaslerUniversity of [email protected]

MS61

Fractional Nonlinear Schroedinger Equations

This talk reviews recent results on fractional nonlinearSchroedinger equations (NLS). This novel class of disper-sive PDEs contains model equations for various physi-cal applications, ranging from relativistic astrophysics tomolecular dynamics. I will present results concerning well-posedness of the initial-value problem, finite-time blowup,and solitary wave solutions for fractional NLS. As a guid-ing example, I will discuss the so-called pseudo-relativisticHartree equation which arises as an effective model equa-tion for relativistic, self-gravitating matter in astrophysics.

Enno [email protected]

MS61

Mathematical Aspects of Bose-Einstein Condensa-tion

The Bose-Einstein condensation (BEC) was predicted in1924 and discovered experimentally in trapped gases in1995. It offers challenging and beautiful mathematicalproblems in Quantum Many-Body Theory and in nonlin-ear PDEs. In the latter case the main questions deal withbehavior of solitons in the nonlinear Schrdinger equationwith an external potential (the Gross-Pitaevskii equation).In this talk I review some recent results and open problemsin the theory of BEC and, specifically, on the dynamics ofsolitons in the Gross-Pitaevskii equations.

Michael SigalUniversity of TorontoToronto, [email protected]

36 PD07 Abstracts - SIAM

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