Analysis of geometry-driven phenomena in fluid mechanics, PDEs and spectral theory

ongoing

This project aims to go significantly beyond the state of the art in several fundamental questions in PDEs with a clear geometric flavor. Central to this proposal is the Euler equation for an incompressible fluid, where the topics that I will be concerned with range from free boundary problems where I will strive to prove that the curvature of the interface blows up in finite time due to the appearance of kinks of controlled geometry, to the existence of smooth stationary solutions that feature chaotic trajectories confined in knotted vortex tubes of any topology, as conjectured by V.I. Arnold over fifty years ago. I will also consider a number of questions in spectral theory about the geometry of the eigenfunctions of the Laplacian and of the curl operator (the so called Beltrami fields, of crucial importance in the study of stationary Euler flows), analyze the process of creation and destruction of vortex structures in the 3D Navier-Stokes and Gross-Pitaevskii equations, consider blowup problems in magnetohydrodynamics, develop global approximation theorems for dispersive equations, and study the limiting measures of a sequence of solutions to the Seiberg-Witten equation. Key for the feasibility of this deep, ambitious project is that these topics are by no means disjoint, so some common themes and fundamental ideas keep coming up in protean forms throughout the research project, and that I have already achieved major results in essentially all the topics covered in the proposal. This includes the proofs of well-known conjectures in fluid mechanics and spectral theory due to Lord Kelvin (1875), V.I. Arnold (1965), S.T. Yau (1993) and M. Berry (2001). The award of a Consolidator Grant would allow me to consolidate both my position as a leader in my fields of interest and the top-level research group on these topics that I am building.