Existence of knotted vortex tubes in steady Euler flows · in the trajectories of solutions to the...

Existence of knotted vortex tubes in steady Euler flows

Daniel Peralta-Salas

Instituto de Ciencias Matematicas, CSIC

deLeonfest, December 2013

D. Peralta-Salas (ICMAT) Knotted vortex tubes in Euler flows December 2013 1 / 16

Definitions

The Euler equation in R3 for an ideal fluid (i.e. inviscid and incompressible):

∂u

∂t+ (u · ∇)u = −∇P , div u = 0


Definitions


∂u

∂t+ (u · ∇)u = −∇P , div u = 0

The unknowns are:

The velocity of the fluid, u(x , t): a non-autonomous vector field in R3.

The pressure P(x , t): a scalar function (determined up to a constant).


Definitions


∂u

∂t+ (u · ∇)u = −∇P , div u = 0

The unknowns are:



In what follows, we will only consider steady solutions u(x) to the Euler equation.Physically, this situation is generally associated with equilibrium configurations.


Definitions


∂u

∂t+ (u · ∇)u = −∇P , div u = 0

The unknowns are:



In what follows, we will only consider steady solutions u(x) to the Euler equation.Physically, this situation is generally associated with equilibrium configurations.

=⇒u × curl u = ∇B , div u = 0

where B := p + 12u

2 is the Bernoulli function.


Definitions (II)

In the spirit of the Lagrangian approach to fluid mechanics, we will be interestedin the trajectories of solutions to the Euler equation: the dynamical systemsviewpoint.


Definitions (II)


More precisely, let us define:

Vorticity: ω := curl u.

Vortex lines: trajectories of the vector field ω (defined through the ODEx = ω(x)).


Definitions (II)


More precisely, let us define:

Vorticity: ω := curl u.

Vortex lines: trajectories of the vector field ω (defined through the ODEx = ω(x)).

The most important object in this talk is:

Definition (Vortex tube)

A vortex tube is a bounded domain in R3 whose boundary is an embedded torusarising as a union of vortex lines (i.e., the vorticity ω is tangent to the boundary).In other words, it is a domain bounded by an invariant torus of ω.


The problem

QuestionAre there any steady solutions to the Euler equation with knotted and linkedvortex tubes?


The problem


knotted tube = smoothly embedded solid torus in R3

linked tubes = union of pairwise disjoint knotted tubes


The problem


knotted tube = smoothly embedded solid torus in R3

linked tubes = union of pairwise disjoint knotted tubes

Figure: Trefoil knot and the Borromean rings.


Motivation

(1) Lord Kelvin’s conjecture (1875): knotted and linked thin vortex tubes canarise in stationary solutions to the Euler equation. Kelvin’s atomic theory modeledatoms as thin knotted vortex tubes of the aether.


Motivation


(2) Kelvin’s circulation theorem: in time-dependent Euler, the vorticity istransported:

∂tω = [ω, u] =⇒ ω(x , t) = (φt,t0 )∗ ω(x , t0) ,

with φt,t0 the non-autonomous flow of u. Hence vorticity evolves through adiffeomorphism. Kelvin’s conjecture is motivated by this phenomenon and by thebelief in the existence of asymptotic states. A more sophisticated variant of thisargument in the context of MHD, called magnetic relaxation, was proposed byMoffatt in the 1980s.


Motivation


(2) Kelvin’s circulation theorem: in time-dependent Euler, the vorticity istransported:

∂tω = [ω, u] =⇒ ω(x , t) = (φt,t0 )∗ ω(x , t0) ,

with φt,t0 the non-autonomous flow of u. Hence vorticity evolves through adiffeomorphism. Kelvin’s conjecture is motivated by this phenomenon and by thebelief in the existence of asymptotic states. A more sophisticated variant of thisargument in the context of MHD, called magnetic relaxation, was proposed byMoffatt in the 1980s.

(3) Linked vortex tubes have been used to construct possible blow-up scenariosfor the Euler equation, e.g. Pelz’s proposal. The idea is that if vortex tubes collideor collapse, this contradicts Kelvin’s circulation theorem, and hence the solutionhas blown up.


Motivation (II)

(4) Kelvin was motivated by Maxwell’s observations of “water twists”. Thinknotted vortex tubes have been recently realized experimentally by Kleckner andIrvine at Chicago:


Motivation (II)


Figure: Trefoil vortex tube in water (Kleckner & Irvine, Nature Phys. 2013)


Motivation (II)


Figure: Trefoil vortex tube in water (Kleckner & Irvine, Nature Phys. 2013)

This vortex tube is not stationary: after some time it reconnects due to theviscosity.


Beltrami fields

To approach this problem we will resort to a particular class of solutions to theEuler equation known as Beltrami fields:

curl u = λu , λ nonzero real constant.


Beltrami fields



It is straightforward to check that any Beltrami field u solves the steady Eulerequation with pressure P = c − 1

2 |u|2 ⇐⇒ constant Bernoulli function.


Beltrami fields





Why Beltrami?:


Beltrami fields





Why Beltrami?:

(1) The Beltrami equation is linear (although not quite elliptic).


Beltrami fields





Why Beltrami?:


(2) Arnold’s structure theorem (1965): under mild hypotheses, the vortex lines ofa steady solution for which u and ω are not everywhere collinear (⇐⇒ nonconstant Bernoulli function) sit over a very rigid integrable structure.


Beltrami fields





Why Beltrami?:


(2) Arnold’s structure theorem (1965): under mild hypotheses, the vortex lines ofa steady solution for which u and ω are not everywhere collinear (⇐⇒ nonconstant Bernoulli function) sit over a very rigid integrable structure.

Non-collinearity of u and ω =⇒ obstructions on knot and link typesArnold’s guess: no obstructions for Beltrami fields.


Beltrami fields (II)

(3) On the contrary, we do not have these problems with Beltrami fields. Moreprecisely:

Theorem (Enciso & P-S, Ann. of Math. 2012)

Given any finite link L, we can deform it with a small diffeomorphism Φ of R3

(close to the identity in the Cm norm) so that Φ(L) is a set of vortex lines of aBeltrami field in R3. These vortex lines are hyperbolic.


Beltrami fields (II)

(3) On the contrary, we do not have these problems with Beltrami fields. Moreprecisely:

Theorem (Enciso & P-S, Ann. of Math. 2012)

Given any finite link L, we can deform it with a small diffeomorphism Φ of R3

(close to the identity in the Cm norm) so that Φ(L) is a set of vortex lines of aBeltrami field in R3. These vortex lines are hyperbolic.

The question now is: do there exist knotted and linked vortex tubes in Beltramifields?


The realization theorem: thin vortex tubes

A convenient way of constructing thin tubes is as metric neighborhoods of curves.




If γ ⊂ R3 is a smooth closed curve, we will denote by

Tε(γ) :={x ∈ R3 : dist(x , γ) < ε

}the thin tube of core γ and thickness ε.





Tε(γ) :={x ∈ R3 : dist(x , γ) < ε


Any finite collection of disjoint tubes in R3 can be isotoped to a collection of thintubes.





Tε(γ) :={x ∈ R3 : dist(x , γ) < ε


Any finite collection of disjoint tubes in R3 can be isotoped to a collection of thintubes.

The realization theorem for thin vortex tubes (Enciso & P-S, preprint)

Let γ1, . . . , γN be N pairwise disjoint (possibly knotted and linked) closed curvesin R3. For small enough ε, one can transform the collection of pairwise disjointthin tubes Tε(γ1), . . . , Tε(γN) by a diffeomorphism Φ of R3, arbitrarily close to theidentity in any C k norm, so that Φ[Tε(γ1)], . . . ,Φ[Tε(γN)] are thin vortex tubes ofa Beltrami field u, which satisfies the equation curl u = λu in R3 for some nonzeroconstant λ. In particular, u is analytic.


The realization theorem: thin vortex tubes (II)

Moreover, the Beltrami field u of the realization theorem has the followingproperties:




(1) It decays at infinity as |D ju(x)| < Cj/|x | for all j . In particular, u ∈ Lp(R3)for p > 3.





(2) There is a positive-measure set of invariant tori close to the boundary of eachvortex tube. On each of these tori, the vorticity flow is ergodic.






(3) There is a closed vortex line near the core curve γi for each i . It is elliptic.







(4) There are infinitely many closed vortex lines in the interior of each vortextube, not of the same knot type as the core vortex line.







(4) There are infinitely many closed vortex lines in the interior of each vortextube, not of the same knot type as the core vortex line.

Corollary

There exist vortex lines of any knot or link type. This recovers our previoustheorem, and improves it because now vortex lines are elliptic.


The realization theorem: thin vortex tubes (III)

Remark 1: The thinness of the tubes is crucially used in the proof of thetheorem: the parameter ε must be small to control certain quantities.




Remark 2: A Beltrami field cannot have finite energy, but the solutions weconstruct have optimal decay in the class of Beltrami fields. The existence ofsteady solutions of 3D Euler with finite energy is open.




Remark 2: A Beltrami field cannot have finite energy, but the solutions weconstruct have optimal decay in the class of Beltrami fields. The existence ofsteady solutions of 3D Euler with finite energy is open.

Remark 3: The vortex tubes are stable in the following sense

1 They are robust under small perturbations of the vector field u.

2 They are stable in a Lyapunov sense, i.e. vortex lines intersecting aneighborhood of a tube remain in a neighborhood.


Some ideas of the proof

The construction of steady solution to the Euler equation with invariant tori is aproblem with both PDE and topological aspects:




Topological techniques are usually too “soft” to capture what happens in aPDE (e.g. Etnyre & Ghrist approach using contact topology).

Purely analytical techniques have been successful in these kind of problemsonly for dimension 2 (e.g. Laurence-Stredulinsky variational approach).




Topological techniques are usually too “soft” to capture what happens in aPDE (e.g. Etnyre & Ghrist approach using contact topology).

Purely analytical techniques have been successful in these kind of problemsonly for dimension 2 (e.g. Laurence-Stredulinsky variational approach).

Strategy1 Construct a local Beltrami field v (that is, in a neighborhood of the tube)

with a prescribed set of vortex tubes (or invariant tori).

2 Check that these invariant tori are robust (that is, they are preserved undersmall perturbations of the field v).

3 Approximate the local solution v by a global Beltrami field u (defined in allR3).


Some ideas of the proof (II)

The construction of the local solution: First, we analyze the harmonic field h of atube

div h = 0 , curl h = 0 , h is tangent to ∂Tεfor small ε. This requires fine anisotropic energy estimates for the Laplacian thatare optimal with respect to ε.





Second, we prove that the harmonic projection h of a Beltrami field v in the tubecan be prescribed, and we estimate their difference in an optimal way:

‖v − h‖Hk 6 Ck |λ| .

Here Ck is a constant that does not depend on ε nor on λ.





Second, we prove that the harmonic projection h of a Beltrami field v in the tubecan be prescribed, and we estimate their difference in an optimal way:

‖v − h‖Hk 6 Ck |λ| .

Here Ck is a constant that does not depend on ε nor on λ.

ConclusionAll the previous carefully established estimates allow to construct a local Beltramifield in Tε that you can control provided that λ and ε are small enough.


Some ideas of the proof (III)

The robustness of the invariant tori: the main tool is KAM theory. The objectthat we use is the Poincare map Π : D2 → D2 of the local Beltrami field v .




The application of KAM theory is not immediate because the Poincare map ishighly degenerate: it is a small perturbation of a twist map with constant rotationnumber:

(r , θ)→ (r , θ +

∫ `

0

τ(α)dα) ,

Here τ is the torsion of the curve. That is why we needed a very careful treatmentof the harmonic field.




The application of KAM theory is not immediate because the Poincare map ishighly degenerate: it is a small perturbation of a twist map with constant rotationnumber:

(r , θ)→ (r , θ +

∫ `

0

τ(α)dα) ,

Here τ is the torsion of the curve. That is why we needed a very careful treatmentof the harmonic field.

Assuming that λ = ε3, after a hard and messy computation, we get the rotationnumber and the normal torsion of Π on the invariant circle, up to order ε2 and ε3,respectively. This allows us to prove a KAM theorem for Beltrami fields in tubes,for generic core curves γ.


Some ideas of the proof (IV)

The global approximation theorem: we prove a Runge-type approximationtheorem for the curl with decay at infinity:

TheoremThere exists a global Beltrami field u which approximates the local Beltrami fieldv in Tε in the C k norm, and which falls off at infinity as |D ju(x)| < Cj/|x |.





To prove this, first we approximate v by a solution w of the Beltrami equation ina large ball BR . This is done using Green’s functions and duality arguments.





To prove this, first we approximate v by a solution w of the Beltrami equation ina large ball BR . This is done using Green’s functions and duality arguments.

Second, we truncate the corresponding Fourier–Bessel series of w to obtain aglobal Beltrami field which falls off at infinity as 1/|x | and is of the form

u =L∑

l=0

l∑m=−l

clm jl(λr)Ylm(θ, ϕ) ,

where (r , θ, ϕ), jj and Ylm are the spherical coordinates in R3, spherical Besselfunctions and spherical harmonics, respectively.


¡Feliz cumpleanos, Manolo!

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Existence of knotted vortex tubes in steady Euler flows · in the trajectories of solutions to the...

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