Research Fellows Directory
Dr Mario Maurelli
University of York
My research is the field of stochastic fluid dynamics and focuses on singular phenomena in two models, namely the two-dimensional stochastic Euler equations and the Kraichnan model of passive scalars.
Fluid dynamic models often exhibit singular phenomena: just to mention one example, the (recently proved) Onsager conjecture, anomalous dissipation of energy in Euler equations. These phenomena are on one side key features of fluids, on the other side often challenging to study. Together with Zdzislaw Brzezniak and other researchers, my purpose is to investigate some of these phenomena, in situations where noise plays a role.
The two-dimensional Euler equations are not fully understood in the case of unbounded vorticity and may exhibit singularities (uniqueness is not known, vortices can collapse). We add a linear transport-type noise to the equation (in vorticity form) and we aim at generalizing the results to the stochastic case. We focus on: a) existence of solution in the unbounded vorticity case; b) particle approximation (the so-called vortex approximation); c) non-white noises, like fractional Brownian noise.
The Kraichnan model of passive scalars is a simplified, linear, yet rich model for turbulence. Here the noise is poorly correlated in space and particles driven by this noise exhibit splitting. The rigorous mathematical analysis of this model has started with the works of Le Jan and Raimond. We aim at a precise investigation of this splitting phenomena and its consequences. We focus on: a) well-posedness and regularization effect of solutions: b) dissipation and long-time behaviour. Finally we put a Kraichnan noise in Euler equations and investigate possible regularization by noise phenomena: the hope is that a regularization effect of the noise may improve the behaviour of the solution, compared to the deterministic case.