Mid-Term Report Summary - INVARIANT (Invariant manifolds in dynamical systems and PDE)
There are many problems in mathematical-physics where the natural objects of study are certain geometric structures associated with vector fields or functions that solve a partial differential equation. Some examples of these structures are the level sets of the gravitatory potential, magnetic lines, stream lines of a fluid, or the nodal sets of a wave function. We call these objects “invariant manifolds” of the solutions to a PDE, in the sense that if the solution is a vector field, then the object is the union of integral curves of the field (e.g. periodic orbits or invariant tori), and if the solution is a function, then the object is the union of level sets of the function. The goal of this project is to study the invariant manifolds in the context of fluid mechanics and elliptic PDEs, where many important problems and conjectures due to Arnold, Kelvin, Ulam, Yau, etc. have been open for years. The difficulty of these questions is related to the interdisciplinary nature of the problem, which requires a high interaction between methods and technique from several areas of mathematics, including PDEs, dynamical systems and differential geometry. The main achievements obtained during the first half of the project are the following. We have proved the existence of Beltrami flows (a particular class of stationary solutions of the Euler equations) on some compact manifolds exhibiting a set of vortex structures of arbitrarily complicated topology. In particular, this proves the 1965 Arnold’s conjecture in the 3-torus, which is the original setting considered by Arnold when he defined the ABC flows. To achieve this result, the powerful tools introduced by Enciso and Peralta-Salas to prove an analogous realization theorem in Euclidean space, are far from enough, so we had to develop a new technique allowing us to transfer the Euclidean result to the compact setting. This new result is a localization lemma which roughly speaking asserts that any Euclidean Beltrami field can be reproduced by Beltrami fields on some compact manifolds in small regions, provided that the energy of the fields is high enough. Apart from filling the gap of previous work that only provided solutions in the Euclidean space, the importance of this result is that it can be used, combined with techniques from dynamical systems and the analysis of PDEs, to establish the existence of global smooth solutions of the 3D Navier-Stokes equations exhibiting a cascade of vortex reconnections (i.e. change of topology in vortex structures) as complicated as one desires. Even though physicists and applied mathematicians have been aware of this phenomenon for a long time, a rigorous proof of reconnections in Navier-Stokes has never been obtained so far. Another important achievement of the project during this period is that we have solved a problem stated by Ulam in 1935 for generic wires. More precisely, we have shown that a generic knotted wire produces a magnetic field (through the Biot-Savart law) with a magnetic line of the same knot type. This is not only an important contribution to Ulam’s problem, but a concrete attempt to develop new ideas to analyze the structure of the magnetic lines created by wires, whose understanding remains strikingly limited despite their manifold applications in engineering. The third remarkable achievement of the project up to now is the solution to a problem posed by Berry in 2001. He conjectured that there should be complex-valued eigenfunctions of the harmonic oscillator in Euclidean space whose nodal set has knotted connected components, and raised the question of whether there can be eigenfunctions of a quantum system whose nodal set has components with higher order linking. Our solution to these problems of Berry consists in showing that any finite link can be realized as a collection of connected components of the nodal set of a high-energy eigenfunction of the harmonic oscillator or the hydrogen atom. Quite unexpected, the techniques to obtain this result are very related to the previously explained localization lemma developed during the study of Beltrami fields on compact manifolds.
María Ángles López Vázquez, (Programme Coordinator)
Record Number: 189119 / Last updated on: 2016-09-20