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Geometric problems in PDEs with applications to fluid mechanics

Periodic Reporting for period 3 - GEOFLUIDS (Geometric problems in PDEs with applications to fluid mechanics)

Reporting period: 2018-03-01 to 2019-08-31

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. 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 project consists of three interrelated parts: the analysis of stationary fluid flows, geometric evolution problems, and global approximation theorems.
The results that have been obtained so far are very satisfactory.

Concerning stationary problems in fluid mechanics, we have just proved a 1965 conjecture of Vladimir Arnold on the existence of chaotic regions in stationary Euler flows (with Luque and Peralta-Salas), as well as a sharp stability theorem for the very important class of stationary fluid flows known as Beltrami fields (with Poyato and Soler). 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 (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).

Concerning global approximation theorems, we are finishing a paper with a complete theory for parabolic equations and decay estimates (with Garcia-Ferrero and Peralta-Salas). With the same authors we have found an application to minimal graphs that deals with intersections between a minimal graph and a horizontal hyperplane.

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) and constructed metrics on any 3-manifold whose associated Laplacian has eigenfunctions with nodal sets of any knot type (with Hartley and Peralta-Salas).

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).
Many of the results that we have derived in this period 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, and a 1993 problem of Yau.