Turbulence is one of the most important mechanisms that govern our environment, affecting air travel,
prediction of ocean surface and weather forecasting. As put in relevance by the 2021 Nobel prize to Giorgio Parisi,
approximate theoretical models and their consequent numeric simulation have been vital to control and predict those
phenomena. Still, a rigorous understanding of the mechanisms that drive turbulence remains one of the main open
problems in mathematical physics. In particular, the rigorous mathematical description of many central properties of
the theory such as intermittency, multifractality and energy redistribution remains underdeveloped. On the other
hand, dispersive PDEs and, in particular, the non-linear Schrödinger equation (NLS) are among the most important
mathematical objects studied in Analysis, and incidentally, they are used to model fluid and wave turbulence. For
instance, NLS has been used to derive a kinetic equation, systematically applied to model the ocean surface. With
this action, I propose to advance in the theories of both turbulence and dispersive PDEs by further exploring the
interaction between them, which will result in a better understanding of the mathematical structure of turbulence. I
propose a Work Package for each direction of this interaction:
Work Package 1: study the concepts of intermittency and multifractality from a rigorous analytic point of view via well-known dispersive models like NLS and the Vortex Filament Equation, which will contribute towards a better understanding of the mathematical theory of turbulence.
Work Package 2: combine classical dispersive PDE and Fourier Analysis problems with probabilistic techniques arising from the study of turbulence, which has the potential of opening a whole new research line. In particular, making use of my previous experience, to study the almost everywhere pointwise convergence problem for NLS and for other dispersive equations from a probabilistic viewpoint.
