Pseudo-Parabolic Partial Differential...
Transcript of Pseudo-Parabolic Partial Differential...
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The Initial-Boundary-Value ProblemsOperators in L2
Pseudo-Parabolic Partial Differential Equations
R.E. Showalter
Department of MathematicsOregon State University
Applied Mathematics & Computation Seminar
RES AMC Seminar 2007
![Page 2: Pseudo-Parabolic Partial Differential Equationssites.science.oregonstate.edu/~show/docs/pseudo... · 2007-03-09 · The Initial-Boundary-Value Problems Operators in L2 Parabolic Diffusion](https://reader034.fdocuments.net/reader034/viewer/2022050418/5f8de8eb8eea4c35c93a4f78/html5/thumbnails/2.jpg)
The Initial-Boundary-Value ProblemsOperators in L2
Outline
1 The Initial-Boundary-Value ProblemsParabolic Diffusion EquationPseudo-Parabolic EquationOrigins
2 Operators in L2
Elliptic Boundary-Value ProblemEvolution Equations in L2(G)ODE and an Elliptic BVP
RES AMC Seminar 2007
![Page 3: Pseudo-Parabolic Partial Differential Equationssites.science.oregonstate.edu/~show/docs/pseudo... · 2007-03-09 · The Initial-Boundary-Value Problems Operators in L2 Parabolic Diffusion](https://reader034.fdocuments.net/reader034/viewer/2022050418/5f8de8eb8eea4c35c93a4f78/html5/thumbnails/3.jpg)
The Initial-Boundary-Value ProblemsOperators in L2
Parabolic Diffusion EquationPseudo-Parabolic EquationOrigins
PDE are just ODE in an appropriate function space.Here we treat simple partial differential equations as evolutionequations (ordinary differential equations) in the space L2(G).
RES AMC Seminar 2007
![Page 4: Pseudo-Parabolic Partial Differential Equationssites.science.oregonstate.edu/~show/docs/pseudo... · 2007-03-09 · The Initial-Boundary-Value Problems Operators in L2 Parabolic Diffusion](https://reader034.fdocuments.net/reader034/viewer/2022050418/5f8de8eb8eea4c35c93a4f78/html5/thumbnails/4.jpg)
The Initial-Boundary-Value ProblemsOperators in L2
Parabolic Diffusion EquationPseudo-Parabolic EquationOrigins
Outline
1 The Initial-Boundary-Value ProblemsParabolic Diffusion EquationPseudo-Parabolic EquationOrigins
2 Operators in L2
Elliptic Boundary-Value ProblemEvolution Equations in L2(G)ODE and an Elliptic BVP
RES AMC Seminar 2007
![Page 5: Pseudo-Parabolic Partial Differential Equationssites.science.oregonstate.edu/~show/docs/pseudo... · 2007-03-09 · The Initial-Boundary-Value Problems Operators in L2 Parabolic Diffusion](https://reader034.fdocuments.net/reader034/viewer/2022050418/5f8de8eb8eea4c35c93a4f78/html5/thumbnails/5.jpg)
The Initial-Boundary-Value ProblemsOperators in L2
Parabolic Diffusion EquationPseudo-Parabolic EquationOrigins
Parabolic equation
u = u(x , t) : Initial-Boundary-Value Problem
∂u∂t−∇·k∇u = 0, x ∈ Ω, t > 0,
u(s, t) = 0, s ∈ ∂Ω, t > 0,
u(x , 0) = u0(x), x ∈ Ω.
RES AMC Seminar 2007
![Page 6: Pseudo-Parabolic Partial Differential Equationssites.science.oregonstate.edu/~show/docs/pseudo... · 2007-03-09 · The Initial-Boundary-Value Problems Operators in L2 Parabolic Diffusion](https://reader034.fdocuments.net/reader034/viewer/2022050418/5f8de8eb8eea4c35c93a4f78/html5/thumbnails/6.jpg)
The Initial-Boundary-Value ProblemsOperators in L2
Parabolic Diffusion EquationPseudo-Parabolic EquationOrigins
Outline
1 The Initial-Boundary-Value ProblemsParabolic Diffusion EquationPseudo-Parabolic EquationOrigins
2 Operators in L2
Elliptic Boundary-Value ProblemEvolution Equations in L2(G)ODE and an Elliptic BVP
RES AMC Seminar 2007
![Page 7: Pseudo-Parabolic Partial Differential Equationssites.science.oregonstate.edu/~show/docs/pseudo... · 2007-03-09 · The Initial-Boundary-Value Problems Operators in L2 Parabolic Diffusion](https://reader034.fdocuments.net/reader034/viewer/2022050418/5f8de8eb8eea4c35c93a4f78/html5/thumbnails/7.jpg)
The Initial-Boundary-Value ProblemsOperators in L2
Parabolic Diffusion EquationPseudo-Parabolic EquationOrigins
Pseudo-Parabolic Equation
∂u∂t− ε∇·k∇∂u
∂t−∇·k∇u = 0, x ∈ Ω, t > 0,
u(s, t) = 0, s ∈ ∂Ω, t > 0,
u(x , 0) = u0(x), x ∈ Ω.
RES AMC Seminar 2007
![Page 8: Pseudo-Parabolic Partial Differential Equationssites.science.oregonstate.edu/~show/docs/pseudo... · 2007-03-09 · The Initial-Boundary-Value Problems Operators in L2 Parabolic Diffusion](https://reader034.fdocuments.net/reader034/viewer/2022050418/5f8de8eb8eea4c35c93a4f78/html5/thumbnails/8.jpg)
The Initial-Boundary-Value ProblemsOperators in L2
Parabolic Diffusion EquationPseudo-Parabolic EquationOrigins
Outline
1 The Initial-Boundary-Value ProblemsParabolic Diffusion EquationPseudo-Parabolic EquationOrigins
2 Operators in L2
Elliptic Boundary-Value ProblemEvolution Equations in L2(G)ODE and an Elliptic BVP
RES AMC Seminar 2007
![Page 9: Pseudo-Parabolic Partial Differential Equationssites.science.oregonstate.edu/~show/docs/pseudo... · 2007-03-09 · The Initial-Boundary-Value Problems Operators in L2 Parabolic Diffusion](https://reader034.fdocuments.net/reader034/viewer/2022050418/5f8de8eb8eea4c35c93a4f78/html5/thumbnails/9.jpg)
The Initial-Boundary-Value ProblemsOperators in L2
Parabolic Diffusion EquationPseudo-Parabolic EquationOrigins
Origins
1926 Milne ... time delay, gas diffusion
1948 Rubinstein ... heat conduction in composite medium
1960 Barenblatt ... fluid flow in fissured medium
1960 Coleman-Noll ... heat conduction
1968 Chen-Gurtin
1966 Lighthill ... fluid
1966 Peregrine ... long waves (semilinear)
1972 Benjamin-Bona-Mahoney
1979 Aifantis ... highly-diffusive paths
1980 Gilbert ... Slightly-compressible Stokes velocity
RES AMC Seminar 2007
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The Initial-Boundary-Value ProblemsOperators in L2
Elliptic Boundary-Value ProblemEvolution Equations in L2(G)
ODE and an Elliptic BVP
Outline
1 The Initial-Boundary-Value ProblemsParabolic Diffusion EquationPseudo-Parabolic EquationOrigins
2 Operators in L2
Elliptic Boundary-Value ProblemEvolution Equations in L2(G)ODE and an Elliptic BVP
RES AMC Seminar 2007
![Page 11: Pseudo-Parabolic Partial Differential Equationssites.science.oregonstate.edu/~show/docs/pseudo... · 2007-03-09 · The Initial-Boundary-Value Problems Operators in L2 Parabolic Diffusion](https://reader034.fdocuments.net/reader034/viewer/2022050418/5f8de8eb8eea4c35c93a4f78/html5/thumbnails/11.jpg)
The Initial-Boundary-Value ProblemsOperators in L2
Elliptic Boundary-Value ProblemEvolution Equations in L2(G)
ODE and an Elliptic BVP
Elliptic Boundary-Value Problem
The spatial derivatives are given by the operator
Au = −∇·k∇u(·) in L2(G),
D(A) = u ∈ H2(G) : u = 0 on ∂G
Eigen-functions: vj(·) : j ≥ 1 is an ortho-normal basis forL2(G)
A(vj) = λjvj , j ≥ 1 , 0 < λj → +∞
RES AMC Seminar 2007
![Page 12: Pseudo-Parabolic Partial Differential Equationssites.science.oregonstate.edu/~show/docs/pseudo... · 2007-03-09 · The Initial-Boundary-Value Problems Operators in L2 Parabolic Diffusion](https://reader034.fdocuments.net/reader034/viewer/2022050418/5f8de8eb8eea4c35c93a4f78/html5/thumbnails/12.jpg)
The Initial-Boundary-Value ProblemsOperators in L2
Elliptic Boundary-Value ProblemEvolution Equations in L2(G)
ODE and an Elliptic BVP
Outline
1 The Initial-Boundary-Value ProblemsParabolic Diffusion EquationPseudo-Parabolic EquationOrigins
2 Operators in L2
Elliptic Boundary-Value ProblemEvolution Equations in L2(G)ODE and an Elliptic BVP
RES AMC Seminar 2007
![Page 13: Pseudo-Parabolic Partial Differential Equationssites.science.oregonstate.edu/~show/docs/pseudo... · 2007-03-09 · The Initial-Boundary-Value Problems Operators in L2 Parabolic Diffusion](https://reader034.fdocuments.net/reader034/viewer/2022050418/5f8de8eb8eea4c35c93a4f78/html5/thumbnails/13.jpg)
The Initial-Boundary-Value ProblemsOperators in L2
Elliptic Boundary-Value ProblemEvolution Equations in L2(G)
ODE and an Elliptic BVP
The Parabolic Equation
u′(t) + Au(t) = 0, t > 0 ,
u(0) = u0 .
u(t) =∞∑
j=1
e−λj t(u0, vj) vj
= S(t)u0 = e−Atu0
Analytic semigroup
Regularity increasing for t > 0
Unbounded decay rate of coefficients
RES AMC Seminar 2007
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The Initial-Boundary-Value ProblemsOperators in L2
Elliptic Boundary-Value ProblemEvolution Equations in L2(G)
ODE and an Elliptic BVP
The Parabolic Equation
u′(t) + Au(t) = 0, t > 0 ,
u(0) = u0 .
u(t) =∞∑
j=1
e−λj t(u0, vj) vj
= S(t)u0 = e−Atu0
Analytic semigroup
Regularity increasing for t > 0
Unbounded decay rate of coefficients
RES AMC Seminar 2007
![Page 15: Pseudo-Parabolic Partial Differential Equationssites.science.oregonstate.edu/~show/docs/pseudo... · 2007-03-09 · The Initial-Boundary-Value Problems Operators in L2 Parabolic Diffusion](https://reader034.fdocuments.net/reader034/viewer/2022050418/5f8de8eb8eea4c35c93a4f78/html5/thumbnails/15.jpg)
The Initial-Boundary-Value ProblemsOperators in L2
Elliptic Boundary-Value ProblemEvolution Equations in L2(G)
ODE and an Elliptic BVP
The Parabolic Equation
u′(t) + Au(t) = 0, t > 0 ,
u(0) = u0 .
u(t) =∞∑
j=1
e−λj t(u0, vj) vj
= S(t)u0 = e−Atu0
Analytic semigroup
Regularity increasing for t > 0
Unbounded decay rate of coefficients
RES AMC Seminar 2007
![Page 16: Pseudo-Parabolic Partial Differential Equationssites.science.oregonstate.edu/~show/docs/pseudo... · 2007-03-09 · The Initial-Boundary-Value Problems Operators in L2 Parabolic Diffusion](https://reader034.fdocuments.net/reader034/viewer/2022050418/5f8de8eb8eea4c35c93a4f78/html5/thumbnails/16.jpg)
The Initial-Boundary-Value ProblemsOperators in L2
Elliptic Boundary-Value ProblemEvolution Equations in L2(G)
ODE and an Elliptic BVP
The Pseudo-Parabolic Equation
u′(t) + εAu′(t) + Au(t) = 0, t > 0 ,
u(0) = u0 .
u(t) =∞∑
j=1
e−λj t
1+ελj (u 0, vj) vj
= Sε(t)u0 = e−(I+εA)−1Atu0
C0-groupRegularity preserving for −∞ < t < ∞
Decay rate bounded below by1ε
RES AMC Seminar 2007
![Page 17: Pseudo-Parabolic Partial Differential Equationssites.science.oregonstate.edu/~show/docs/pseudo... · 2007-03-09 · The Initial-Boundary-Value Problems Operators in L2 Parabolic Diffusion](https://reader034.fdocuments.net/reader034/viewer/2022050418/5f8de8eb8eea4c35c93a4f78/html5/thumbnails/17.jpg)
The Initial-Boundary-Value ProblemsOperators in L2
Elliptic Boundary-Value ProblemEvolution Equations in L2(G)
ODE and an Elliptic BVP
The Pseudo-Parabolic Equation
u′(t) + εAu′(t) + Au(t) = 0, t > 0 ,
u(0) = u0 .
u(t) =∞∑
j=1
e−λj t
1+ελj (u 0, vj) vj
= Sε(t)u0 = e−(I+εA)−1Atu0
C0-groupRegularity preserving for −∞ < t < ∞
Decay rate bounded below by1ε
RES AMC Seminar 2007
![Page 18: Pseudo-Parabolic Partial Differential Equationssites.science.oregonstate.edu/~show/docs/pseudo... · 2007-03-09 · The Initial-Boundary-Value Problems Operators in L2 Parabolic Diffusion](https://reader034.fdocuments.net/reader034/viewer/2022050418/5f8de8eb8eea4c35c93a4f78/html5/thumbnails/18.jpg)
The Initial-Boundary-Value ProblemsOperators in L2
Elliptic Boundary-Value ProblemEvolution Equations in L2(G)
ODE and an Elliptic BVP
The Pseudo-Parabolic Equation
u′(t) + εAu′(t) + Au(t) = 0, t > 0 ,
u(0) = u0 .
u(t) =∞∑
j=1
e−λj t
1+ελj (u 0, vj) vj
= Sε(t)u0 = e−(I+εA)−1Atu0
C0-groupRegularity preserving for −∞ < t < ∞
Decay rate bounded below by1ε
RES AMC Seminar 2007
![Page 19: Pseudo-Parabolic Partial Differential Equationssites.science.oregonstate.edu/~show/docs/pseudo... · 2007-03-09 · The Initial-Boundary-Value Problems Operators in L2 Parabolic Diffusion](https://reader034.fdocuments.net/reader034/viewer/2022050418/5f8de8eb8eea4c35c93a4f78/html5/thumbnails/19.jpg)
The Initial-Boundary-Value ProblemsOperators in L2
Elliptic Boundary-Value ProblemEvolution Equations in L2(G)
ODE and an Elliptic BVP
Outline
1 The Initial-Boundary-Value ProblemsParabolic Diffusion EquationPseudo-Parabolic EquationOrigins
2 Operators in L2
Elliptic Boundary-Value ProblemEvolution Equations in L2(G)ODE and an Elliptic BVP
RES AMC Seminar 2007
![Page 20: Pseudo-Parabolic Partial Differential Equationssites.science.oregonstate.edu/~show/docs/pseudo... · 2007-03-09 · The Initial-Boundary-Value Problems Operators in L2 Parabolic Diffusion](https://reader034.fdocuments.net/reader034/viewer/2022050418/5f8de8eb8eea4c35c93a4f78/html5/thumbnails/20.jpg)
The Initial-Boundary-Value ProblemsOperators in L2
Elliptic Boundary-Value ProblemEvolution Equations in L2(G)
ODE and an Elliptic BVP
ODE in L2(G)
Aε = (I + εA)−1A = A(I + εA)−1 =1ε(I − (I + εA)−1)
is a bounded operator on L2(G).
u′(t) + Aεu(t) = 0
is an Ordinary Differential Equation in L2(G).
RES AMC Seminar 2007
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The Initial-Boundary-Value ProblemsOperators in L2
Elliptic Boundary-Value ProblemEvolution Equations in L2(G)
ODE and an Elliptic BVP
... a little algebra ...
The pseudo-parabolic equation
u′(t) + εAu′(t) + Au(t) = 0, t > 0
can be written
u′(t) +1ε
u(t) =1ε(I + εA)−1u(t) ∈ D(A)
The saltus or jump along an interface, [u](t), satisfies
[u]′(t) +1ε[u](t) = 0 ,
so
[u](t) = e−tε [u0] .
RES AMC Seminar 2007
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The Initial-Boundary-Value ProblemsOperators in L2
Elliptic Boundary-Value ProblemEvolution Equations in L2(G)
ODE and an Elliptic BVP
... a little algebra ...
The pseudo-parabolic equation
u′(t) + εAu′(t) + Au(t) = 0, t > 0
can be written
u′(t) +1ε
u(t) =1ε(I + εA)−1u(t) ∈ D(A)
The saltus or jump along an interface, [u](t), satisfies
[u]′(t) +1ε[u](t) = 0 ,
so
[u](t) = e−tε [u0] .
RES AMC Seminar 2007
![Page 23: Pseudo-Parabolic Partial Differential Equationssites.science.oregonstate.edu/~show/docs/pseudo... · 2007-03-09 · The Initial-Boundary-Value Problems Operators in L2 Parabolic Diffusion](https://reader034.fdocuments.net/reader034/viewer/2022050418/5f8de8eb8eea4c35c93a4f78/html5/thumbnails/23.jpg)
The Initial-Boundary-Value ProblemsOperators in L2
Elliptic Boundary-Value ProblemEvolution Equations in L2(G)
ODE and an Elliptic BVP
Richard’s equation(with M. Peszynska, S.-Y.Yi)
φ∂S∂t
+∇ · (Kkw (S)∇Pc(S)) = ∇ · (Kkw (S)Gρw∇Depth(x))
Rewritten in a generic nonlinear parabolic form
∂S∂t−∇ · (D(S)∇S) = ∇ · (Λ(S))
D(S) is non-negative definite and degenerate
D(S) ≈ 0, S1 ≤ S ≤ S2
Λ(S) is monotone increasing degenerate.
RES AMC Seminar 2007
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The Initial-Boundary-Value ProblemsOperators in L2
Elliptic Boundary-Value ProblemEvolution Equations in L2(G)
ODE and an Elliptic BVP
Richard’s equation with dynamic capillary pressure(with M. Peszynska, S.-Y.Yi)
Replace Pc(S) by Pc(S, ∂S∂t ) to account for dependence on time
scale of getting to capillary equilibrium [Wildenschild et al]
φ∂S∂t
+∇ · (Kkw (S)∇Pc(S,∂S∂t
)) = ∇ · (Kkw (S)Gρw∇Depth(x))
([B]) [Barenblatt] Pc(S, ∂S∂t )) := Pc(S + τ ∂S
∂t )
([HC]) [Hassanizadeh, Celia] Pc(S, ∂S∂t )) := Pc(S)− τ ∂S
∂t(also note some hysteresis models [Belyaev, Schotting, vanDuijn]Can be rewritten in a generic nonlinear pseudo–parabolic form
∂S∂t−∇ · (D(S)∇S) = ∇ · (Λ(S)) +∇ ·
(C(S)∇∂S
∂t
)where C(S) is more or less degenerate depending on themodel RES AMC Seminar 2007
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The Initial-Boundary-Value ProblemsOperators in L2
Elliptic Boundary-Value ProblemEvolution Equations in L2(G)
ODE and an Elliptic BVP
Compare solutions to linear parabolic andpseudo-parabolic equations (M. Peszynska, S.-Y.Yi)
∂S∂t−∇ · (D∇S) = ∇ · C∇(
∂S∂t
)
where C = τD.
use τ = 0 and τ = 1.
initial data: smooth (optimal convergence)
τ = 0 τ = 1RES AMC Seminar 2007
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The Initial-Boundary-Value ProblemsOperators in L2
Elliptic Boundary-Value ProblemEvolution Equations in L2(G)
ODE and an Elliptic BVP
Compare solutions to linear parabolic andpseudo-parabolic equations (M. Peszynska, S.-Y.Yi)
∂S∂t−∇ · (D∇S) = ∇ · C∇(
∂S∂t
)
where C = τD.use τ = 0 and τ = 1.initial data: nonsmooth
nonsmooth initial data, τ = 0 nonsmooth initial data, τ = 1
RES AMC Seminar 2007
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The Initial-Boundary-Value ProblemsOperators in L2
Elliptic Boundary-Value ProblemEvolution Equations in L2(G)
ODE and an Elliptic BVP
RES AMC Seminar 2007