Knots and links in fluid mechanics - UAB Barcelona · [\Knots and links in steady solutions of the Euler equation", arXiv:1003.3122 (Ann. of Math., in press)] Daniel Peralta-Salas

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Knots and links in fluid mechanics

Daniel Peralta-Salas

Instituto de Ciencias Matematicas, CSIC

September 2011

Daniel Peralta-Salas (CSIC) Knots in fluid mechanics September 2011 1 / 5

Statement of the problem

∂u

∂t+ (u · ∇)u = −∇P , div u = 0

This is the Euler equation in R3. Unknowns: u (velocity field), P (pressure).The trajectories of the vector field u are called stream lines, and those of its curlω := curl u (the vorticity) are the vortex lines.

QuestionThe viewpoint of dynamical systems: what do stream and vortex lines look like?Here the stream lines are trajectories of fluid particles (Lagrangian approach)



∂u

∂t+ (u · ∇)u = −∇P , div u = 0

This is the Euler equation in R3. Unknowns: u (velocity field), P (pressure).The trajectories of the vector field u are called stream lines, and those of its curlω := curl u (the vorticity) are the vortex lines.

QuestionWhich knots and links can be stream or vortex lines of a steady solution of theEuler equation? (This is one of the key problems in topological hydrodynamics.)

(steady = u does not depend on t)(knot = embedded circle in R3)(link = union of pairwise disjoint knots)



Figure: Trefoil knot and the Borromean rings.


Motivation

For decades, the conjecture has been that “all” knot and link types should berealizable as stream or vortex lines of a solution to the steady Euler equation. Thisconjecture was popularized by the works of Arnold and Moffatt in the 60’s.

Remarkably, this conjecture is motivated by physical phenomena: magneticrelaxation and transport of vorticity. Indeed, ∂tω = [u, ω], so that

ω(x , t) = (φt,t0 )∗ ω(x , t0) ,

with (φt,t0 )∗ the push-forward of the non-autonomous flow generated by u.

In turbulence, the (conjectured) existence of highly entangled trajectories wasinterpreted as an indication of turbulent behavior.

In view of the general belief in the validity of the conjecture, Arnold’s structuretheorem is somehow surprising, as it asserts that under mild hypotheses streamlines are nicely stacked in a structure akin to that of an integrable Hamiltoniansystem:


Motivation

Theorem (Arnold’s structure theorem, 1965)

Let u be a steady solution of the Euler equation in a smooth bounded domainΩ ⊂ R3, tangent to ∂Ω. Suppose that u is analytic in Ω and that u andω := curl u are not everywhere collinear (u ∧ ω 6≡ 0).

Then there is an analytic set C (with codim C > 1) such that Ω\C consists of afinite number of subdomains, and in each one the dynamics of u is of one of thefollowing forms:

1 The subdomain is trivially fibered by tori invariant under u. In each torus, thestream lines are C∞-conjugate to those of a linear (rational or irrational) flow.

2 The subdomain is trivially fibered by cylinders invariant under u whoseboundaries lie on ∂Ω. In each cylinder, the stream lines are periodic.

Key of the proof: The steady Euler equation can be written as u ∧ ω = ∇B, withB the Bernoulli function. Hence u · ∇B = 0 and [u, ω] = 0.


Motivation

Theorem (Arnold’s structure theorem, 1965)

Let u be a steady solution of the Euler equation in a smooth bounded domainΩ ⊂ R3, tangent to ∂Ω. Suppose that u is analytic in Ω and that u andω := curl u are not everywhere collinear (u ∧ ω 6≡ 0).

Then there is an analytic set C (with codim C > 1) such that Ω\C consists of afinite number of subdomains, and in each one the dynamics of u is of one of thefollowing forms:

1 The subdomain is trivially fibered by tori invariant under u. In each torus, thestream lines are C∞-conjugate to those of a linear (rational or irrational) flow.

2 The subdomain is trivially fibered by cylinders invariant under u whoseboundaries lie on ∂Ω. In each cylinder, the stream lines are periodic.

Arnold: “Boundedness and analyticity should not be essential. Collinearityshould” (very true).


Motivation

Some comments:

1 It is apparent that this rigid structure should lead to obstructions onadmissible knot and link types, under minor assumptions (Etnyre–Ghrist,1999).

2 There is no converse to Arnold’s theorem: given a structure compatible withArnold’s theorem, one does not know if there is a steady solution of the Eulerequation realizing it!


Beltrami fields

In 1965 Arnold conjectured that, in order to construct solutions with acomplicated orbit structure, we should resort to solutions for whichcurl u = f (x) u, for some function f . Since f would be a first integral, it is thennatural to consider fields such that

curl u = λu ,

with λ ∈ R. These are called Beltrami fields, and automatically satisfy the steadyEuler equation with pressure P = − 1

2 |u|2 + const. (These fields also arise in

magnetohydrodynamics, where they are called force-free fields.)

Realization Theorem (Enciso & Peralta-Salas, 2011)

Let L ⊂ R3 be a locally finite link. For any λ 6= 0, we can transform L with adiffeomorphism Φ of R3, close to the identity in any C p norm, such that Φ(L) is aset of periodic stream lines of a solution of the equation curl u = λu in R3.

[“Knots and links in steady solutions of the Euler equation”, arXiv:1003.3122(Ann. of Math., in press)]


Beltrami fields

A locally finite (but scary enough) link. Up to a small diffeomorphism, there is asteady solution of the Euler equation having all these periodic orbits!


Beltrami fields

Corollary (Question by Etnyre & Ghrist)

There exists a steady solution of the Euler equation having periodic orbits of allknot types at the same time.

However, a Beltrami field satisfies ∆u = −λ2u, so the solutions considered in ourtheorem do not have finite energy (u 6∈ L2). It would be very interesting, bothphysically and mathematically, to prove an analog of this result for solutions:

1 in R3 with some decay at infinity.

2 in a bounded domain with tangency boundary conditions.

3 in the flat 3-torus.


Sketch of the proof

The heart of the problem is that one needs to extract topological information froma PDE. Generally speaking, topological techniques are too “soft” to capture whathappens in a PDE, while analytical techniques have not been very successful inthese kind of problems either.

We used an intermediate, or mixed, approach. The basic philosophy is to use themethods of differential topology and dynamical systems to control auxiliaryconstructions and those of PDEs to relate these auxiliary constructions to theEuler equation.


Sketch of the proof

1 Construction of an auxiliary vector field w1 in a neighborhood of a linkcomponent L1.

Defined in a “strip” Σ (small cylinder) around L1. Its dual 1-form is closed.The field w1 is tangent to the strip and L1 is a stable hyperbolic limit cycle toits pullback to Σ.

2 Local solution v1 of the equation curl v1 = λv1 in a neighborhood of L1 withinitial datum v1|Σ = w1.

CK-type theorem, proved using the Dirac-type operator d + d∗ on Ω•(R3).The vector field used as Cauchy data must be dual to a closed 1-form for asolution to exist.The condition div v1 = 0 ensures that L1 is a hyperbolic periodic trajectory.We repeat the process with all the components, getting a “robust” localsolution v in a small neighborhood of L.

3 Finally we prove that the local solution v can be approximated (together withits first p derivatives) by a global solution u.

As the link L can be infinite, we need a C p better-than-uniform approximation:∑|α|6p

∣∣Dαu(x) − Dαv(x)∣∣ < ε(x)

for any positive continuous function ε(x). By the robustness of hyperboliccycles, the result then follows.


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Knots and links in fluid mechanics - UAB Barcelona · [\Knots and links in steady solutions of the Euler equation", arXiv:1003.3122 (Ann. of Math., in press)] Daniel Peralta-Salas