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

We establish well-posedness in law for a general class of stochastic 2D fluid dynamics equations with $(L^1_x\cap L^2_x)$-valued vorticity and finite kinetic energy; the noise is of Kraichnan type and spatially rough, 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 to be ill-posed; consequently, the addition of noise truly improves the solution theory for such PDE.