Weak well-posedness by transport noise for a class of 2D fluid dynamics equations

Abstract: A fundamental open problem in fluid dynamics is whether solutions to $2$D Euler equations with $(L^1_x\cap L^p_x)$-valued vorticity are unique, for some $p\in [1,\infty)$. A related question, more probabilistic in flavour, is whether one can find a physically relevant noise regularizing the PDE. We present some substantial advances towards a resolution of the latter, by establishing well-posedness in law for solutions with $(L^1_x\cap L^2_x)$-valued vorticity and finite kinetic energy, for a general class of stochastic 2D fluid dynamical equations; the noise is spatially rough and of Kraichnan type and we allow the presence of a deterministic forcing $f$. This class includes as primary examples logarithmically regularized 2D Euler and hypodissipative 2D Navier-Stokes equations. In the first case, our result solves the open problem posed by Flandoli. In the latter case, for well-chosen forcing $f$, the corresponding deterministic PDE without noise has recently been shown by Albritton and Colombo to be ill-posed; consequently, the addition of noise truly improves the solution theory for such PDE.