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Geometric problems in PDEs with applications to fluid mechanics

Periodic Reporting for period 4 - GEOFLUIDS (Geometric problems in PDEs with applications to fluid mechanics)

Reporting period: 2019-09-01 to 2021-02-28

In this project we are developing innovative new tools to understand a wide range of geometric problems appearing in differential equations, particularly in fluid mechanics. Our goal is to develop ideas and techniques to be able to rigorously analyze the phenomena described by these equations, which pertain to manifold fields such as fluid mechanics, quantum physics and relativity. This is basic research aimed at re-sahping the way in which we understand the mathematics underlying many important physical effects, such as turbulence.

The basic philosophy of the project leads to significant advances in a number of apparently unrelated questions that are nevertheless connected through mathematical concepts. Specifically, the project consists of three interrelated parts: the analysis of stationary fluid flows, geometric evolution problems, and global approximation theorems. In all three blocks we explore fundamental questions in PDE that are often motivated by the study of physical phenomena.

From the point of view of mathematics, this is both very challenging and extremely rewarding. Indeed, the difficulty is apparent in view of the number of high-profile questions in the field that go back to the leading figures of contemporary mathematics and which have remained open for decades. This is mostly due to the fact that the highly interdisciplinary nature of these problems requires ingenious combinations of ideas from many areas of mathematics to make progress: analysis, differential geometry, topology and dynamical systems go hand in hand here. The major upside of this difficulty, of course, is that significant contributions in this direction attract considerable attention as they often lead to deep, unexpected connections between different areas of mathematics.
The results that have been obtained are very satisfactory. In shall next mention some of these results:

Concerning stationary problems in fluid mechanics, we have established the generic existence of knotted vortex structures and chaotic regions in Beltrami fields, in perfect agreement with Arnold's vision of stationary Euler flows (with Luque, Peralta-Salas and Romaniega), as well as a sharp stability theorem for Beltrami fields (with Poyato and Soler). With Luque and Peralta-Salas, we have found new ways to analyze the splitting of separatrices in certain mechanical problems with fast oscillations. We have also contracted knotted structures in fluids with periodic boundary conditions (with Peralta-Salas and Torres de Lizaur) and shown with the same coauthors that the helicity is the only integral invariant for volume-preserving diffeomorphisms, thereby answering a 1998 question of Arnold and Khesin. We have also constructed stationary splash singularities, which can in fact pinch an incompressible fluid (with Cordoba and Grubic). In the 3D Navier-Stokes equations we have constructed the first rigorous scenario of vortex reconnection, which is a major problem (with Luca and Peralta-Salas). We also studied the Biot-Savart kernel associated with a bounded domain (with Garcia-Ferrero and Peralta-Salas).

Concerning global approximation theorems, we have developed a complete theory for parabolic equations (with Garcia-Ferrero and Peralta-Salas) and for the Schrödinger equation (with Peralta-Salas). We have also considered applications of these ideas to the study of reconnections in Gross-Pitaevskii, to the movement of local hot spots of parabolic equations and to the construction of minimal graphs with micro-oscillations.

In spectral theory we have answered a 1993 question of Yau about nodal sets of low-lying eigenvalues on manifolds (with Peralta-Salas), even when one only allows for conformal deformations of the metric (with Peralta-Salas and Steinerberger). We have also answered a 2001 question of Berry on knotted nodal sets in quantum mechanics (with Hartley and Peralta-Salas), constructed metrics on any 3-manifold whose associated Laplacian has eigenfunctions with nodal sets of any knot type (with Hartley and Peralta-Salas) and proved that regular polygons are spectrally determined (with Gomez-Serrano).

Rather unexpectedly, we have made an important contribution to a 1935 question of Ulam about the integral curves of vector fields generated by knotted wires, which in particular solves it in the generic case (with Peralta-Salas).

In wave equations, we have proved that there are Lorentzian Einstein metrics with prescribed conformal infinity (with Kamran), which is a Lorenztian analog of a classical result of Graham-Lee that applies to spacetimes close to anti-de Sitter. We have also found a connection between the Dirichlet-Neumann map for waves on anti-de Sitter spaces and fractional powers of the wave operator (with Gonzalez and Vergara). We also obtained sharp global Carleman estimates for wave equations with potentials diverging as the inverse square of the distance to a hypersurface, which are closely related to waves on AdS backgrounds (with Shao and Vergara).

Many of these results have been disseminated through invited and plenary talks in conferences and through seminars and colloquia. Some of them have also been presented in the media, and discussed in various round tables and science dissemination events.
Many of the results that we have obtained, described above, go well beyond the state of the art and have opened up new perspectives that I plan to study in detail in the following years. This is evidenced by the solution of a 1998 question of Arnold-Khesin, a 2001 conjecture of Sir Michael Berry, a 1993 problem of Yau, and the proof that generic Beltrami fields exhibit knotted structures and chaotic behavior.
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