Periodic Reporting for period 2 - FLUSPEC (Analysis of geometry-driven phenomena in fluid mechanics, PDEs and spectral theory)
Período documentado: 2022-09-01 hasta 2024-02-29
The very formulation of many high-profile problems in PDEs, some of which go back to the leading mathematicians of the last century, entails objects such as curves, level sets, critical points, cusps or corners, which ultimately makes them of a strongly geometric nature. It is now folk wisdom that, despite the intrinsic complexity of this kind of problems, their dual analytic and geometric nature can be a blessing in disguise, for solutions to these problems often open up new vistas in very different areas of mathematics. In a way, unveiling some of the underlying connections between different areas of mathematics is the most fundamental objective of this project.
This project aims to go significantly beyond the state of the art in several fundamental questions in PDEs with a clear geometric flavor. We will consider three kinds of problems. First, those related to singular (that is, low regularity) solutions in incompressible fluid mechanics. In this setting, we are mostly interested in the 3D Euler and MHD equations, with or without a free boundary, although some 1D and 2D models also provide extremely interesting models for the expected three-dimensional behavior. Second, we are interested in smooth phenomena in PDE related to qualitative changes of topology. We will explore these questions in the setting of the 3D Euler equations (where, in particular, we want to analyze the emergence of chaotic patterns in fluid in equilibrium), of nonlinear Schrödinger equations, and in Taubes' system of modified Seiberg-Witten equations. The third kind of problems we will consider concern random and deterministic questions in spectral theory, and particularly the phenomenon of inverse localization for high energy eigenfunctions.
This project aims to go significantly beyond the state of the art in several fundamental questions in PDEs with a clear geometric flavor. We will consider three kinds of problems. First, those related to singular (that is, low regularity) solutions in incompressible fluid mechanics. In this setting, we are mostly interested in the 3D Euler and MHD equations, with or without a free boundary, although some 1D and 2D models also provide extremely interesting models for the expected three-dimensional behavior. Second, we are interested in smooth phenomena in PDE related to qualitative changes of topology. We will explore these questions in the setting of the 3D Euler equations (where, in particular, we want to analyze the emergence of chaotic patterns in fluid in equilibrium), of nonlinear Schrödinger equations, and in Taubes' system of modified Seiberg-Witten equations. The third kind of problems we will consider concern random and deterministic questions in spectral theory, and particularly the phenomenon of inverse localization for high energy eigenfunctions.
In the first block, we have already obtained good results about free-boundary Euler with interfaces that feature cusps and corners, and we have constructed a wealth of discontinuous (but piecewise smooth) solutions to the 3D MHD equations by means of a novel strategy. Also in the context of incompressible plasmas, we have derived the obstructions to topological reconnection, which show that generic magnetic fields cannot relax to a stationary state of the same topology. We also have some extremely promising preliminary results on fine uniqueness properties of cusped interfaces for the Whitham equation.
In the second block, we have already obtained very strong results on generic chaos for 3D Euler, and completely satisfactory results about compactly supported Euler flows, quasiperiodic Euler flows and vortex reconnection for the Gross-Pitaevskii equation. We have also obtained a very nice result about limiting measures for Taubes' Seiberg-Witten system.
In the third block, we have analyzed in detail the expected number of critical points for random waves with non-identical variables, which allow for arbitrary almost sure Sobolev regularity of their densities, derived a satisfactory theory of random Beltrami fields, which was actually instrumental both in analyzing chaos in 3D Euler and which has enabled us to show that natural Hamiltonian systems are typically chaotic, in a probabilistic sense. We have also proved a beautiful result about inverse localization in flat tori with arbitrary lattices. We have also considered spectral problems for Beltrami fields, obtaining interesting results on optimal domains and optimal metrics. We also obtained nontrivial solutions to a weaker analog of the celebrated Schiffer problem and, continuing the main idea of using spectral results to advance our studies of incompressible fluids, we have utilized this result to construct compactly supported stationary Euler flows in two dimensions that are not locally radial.
In the second block, we have already obtained very strong results on generic chaos for 3D Euler, and completely satisfactory results about compactly supported Euler flows, quasiperiodic Euler flows and vortex reconnection for the Gross-Pitaevskii equation. We have also obtained a very nice result about limiting measures for Taubes' Seiberg-Witten system.
In the third block, we have analyzed in detail the expected number of critical points for random waves with non-identical variables, which allow for arbitrary almost sure Sobolev regularity of their densities, derived a satisfactory theory of random Beltrami fields, which was actually instrumental both in analyzing chaos in 3D Euler and which has enabled us to show that natural Hamiltonian systems are typically chaotic, in a probabilistic sense. We have also proved a beautiful result about inverse localization in flat tori with arbitrary lattices. We have also considered spectral problems for Beltrami fields, obtaining interesting results on optimal domains and optimal metrics. We also obtained nontrivial solutions to a weaker analog of the celebrated Schiffer problem and, continuing the main idea of using spectral results to advance our studies of incompressible fluids, we have utilized this result to construct compactly supported stationary Euler flows in two dimensions that are not locally radial.
The progress we have obtained is better than we have initially expected, so we should have extremely satisfactory results by the end of the 5-year period.