Turbulence and Interactions with Dispersive Equations and Fourier Analysis

ended

The turbulence of fluids and waves is one of the most important mechanisms that govern our environment, affecting air travel, prediction of ocean surface or weather forecasting. Even if simulations are used every day to predict turbulence, its accurate mathematical description is one of the biggest open problems in Mathematical Physics. On the other hand, dispersive PDEs are central objects in Mathematical Analysis and are among the most successful models for several physical phenomena. In this project I propose to advance in both theories, turbulence and dispersive PDEs, by studying their interaction. The project is divided in two Work Packages, each focused on one direction of such interaction: WP1) To advance in the rigorous mathematical treatment of intermittency and multifractality, central concepts in turbulence. The approach proposed is to identify them in the Vortex Filament Equation, the 1D Schrödinger map on the sphere and the non-linear Schrödinger equation (NLS), well-established models connected through the Hasimoto transformation and directly related to fluid and wave turbulence respectively. This is a novelty with respect to previous works, which mainly focus on abstract, often isolated mathematical objects. WP2) To study the pointwise convergence problem to the initial datum, one of the most important problems in dispersive PDEs and Fourier Analysis, from a novel probabilistic approach coming from fluid and wave turbulence. This is motivated by recent progress in wave turbulence, which has put forth the value of probabilistic techniques. To solve the problems, techniques from Fourier Analysis, PDEs and probability, complemented by number theory and geometry, will be required. For this I will have the support and training of two co-supervisors, experts in complementary areas in Harmonic Analysis and PDEs. Besides, the project will benefit from interdisciplinary collaboration with physician experts in turbulence.