Stochastic Analysis Seminar – Lucio Galeati (University of L’Aquila)

Motivated by turbulence modelling, we consider a class of 2D stochastic PDEs, characterized by the presence of a Gaussian, rough, solenoidal velocity field acting as a transport term. The linear version of such an equation was proposed in the 60s by Kraichnan as a synthetic model for passive scalar turbulence; it is by now well-understood that it displays numerous interesting features, like anomalous dissipation of energy and Lagrangian spontaneous stochasticity. Here we consider more complicated cases of active scalar equations, like 2D Euler in vorticity form and the generalized Surface Quasi Geostrophic (gSQG) equations. When the Gaussian field is white in time and $\alpha$-Holder in space, for some $\alpha\in (0,1)$, the associated transport term has a dissipative nature that helps stabilize the dynamics. As a consequence, for any $\alpha\in (0,1)$ and suitable $p\in (1,\infty)$, we can show strong existence and pathwise uniqueness of global solutions in $L^p$ spaces, which are recovered as the unique limit of vanishing viscosity approximations. This is in sharp contrast with the deterministic system, where such a result is open or even false in the presence of suitable forcing. The proofs are based on suitable refinements of the techniques first introduced in Coghi, Maurelli (2023).
Based on ongoing joint work with M. Bagnara.