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

A fundamental open problem in fluid dynamics is whether solutions to 2D Euler equations with (L^1_x∩ L^p_x)-valued vorticity are unique, for some p∈ [1,∞). 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∩ 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.