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

TL;DR

This paper demonstrates that spatially rough Kraichnan-type noise can regularize certain 2D fluid equations, establishing well-posedness in law for solutions with specific vorticity and energy conditions, including cases where deterministic versions are ill-posed.

Contribution

It proves well-posedness in law for stochastic 2D fluid equations with rough noise, addressing open problems and showing noise can improve solution theory for ill-posed deterministic PDEs.

Findings

Established well-posedness in law for stochastic 2D Euler and Navier-Stokes equations.

Showed noise regularizes PDEs that are ill-posed without stochastic terms.

Included examples of logarithmically regularized equations and hypodissipative models.

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.