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

Abstract

We establish well-posedness in law for a general class of stochastic 2D fluid dynamics equations with $(L_x^1\cap L_x^2)$-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 [43]. In the latter case, for well-chosen forcing f, the corresponding deterministic PDE without noise has recently been shown in [3] to be ill-posed; consequently, the addition of noise truly improves the solution theory for such PDE.