A major open problem in fluid dynamics is to understand whether solutions to 2D incompressible Euler equations with $L^p$-valued vorticity are unique, for some $p\in [1,\infty)$. A related question, more probabilistic in flavour, is whether one can find a physically meaningful noise restoring well-posedness of the PDE.

In this talk I will present some recent advances on the latter problem, for a class of slightly regularised 2D Euler-type equations (specifically, logEuler and hypodissipative Navier-Stokes), in the presence of a rough Kraichnan-type noise, modelling the small scales of a turbulent fluid; uniqueness in law can then be shown for solutions with $L^2$-valued vorticity.

Based on an ongoing joint work with Dejun Luo (Beijing).