Knots and links in fluid mechanics
Seminar Room 1, Newton Institute
In this talk I will discuss the existence of steady solutions to the incompressible Euler equations that have stream and vortex lines of any prescribed knot (or link) type. More precisely, I will show that, given any locally finite link L in R^3, one can transform it by a smooth diffeomorphism F, close to the identity in any C^p norm, such that F(L) is a set of periodic trajectories of a real analytic steady solution u of the Euler equations in R^3. If the link is finite, we shall also see that u can be assumed to decay as 1/|x| at infinity, so that u is in L^p for all p>3. This problem is motivated by the well-known analysis of the structure of steady incompressible flows due to V.I. Arnold and K. Moffatt, among others.
Time permitting, we will also very recent results on the topology of potential flows, that is, of steady fluids whose velocity field is the gradient of a harmonic function in R^3. These results are closely related to classic questions in potential theory that were first considered by M. Morse and W. Kaplan in the first half of the XX century and have been revisited several times after that, by Rubel, Shiota and others.
The guiding principle of the talk will be that a strategy of "local, analysis-based constructions" + "global approximation methods", fitted together using ideas from differential topology, can be used to shed some light on the qualitative behavior of steady fluid flows. Most of the original results presented in this talk will be based on the papers:
A. Enciso, D. Peralta-Salas, Knots and links in steady solutions of the Euler equation, Ann. of Math. 175 (2012) 345-367.
A. Enciso, D. Peralta-Salas, Submanifolds that are level sets of solutions to a second-order elliptic PDE, arXiv:1007.5181.
A. Enciso, D. Peralta-Salas, Arnold's structure theorem revisited, in preparation.