Existence and uniqueness by Kraichnan noise for 2D Euler equations with unbounded vorticity

TL;DR

This paper proves existence and uniqueness of solutions for 2D Euler equations with unbounded vorticity under Kraichnan noise, showing noise can regularize and stabilize solutions in a stochastic setting.

Contribution

It establishes weak existence and pathwise uniqueness for stochastic 2D Euler equations with unbounded vorticity, a novel result enabled by Kraichnan noise.

Findings

Weak existence for unbounded initial vorticity.

Solutions are limits of regularized stochastic Euler equations.

Pathwise uniqueness for certain initial vorticities with noise.

Abstract

We consider the 2D Euler equations on $\R^2$ in vorticity form, with unbounded initial vorticity, perturbed by a suitable non-smooth Kraichnan transport noise, with regularity index $\alpha\in (0,1)$. We show weak existence for every $\dot{H}^{-1}$ initial vorticity. Thanks to the noise, the solutions that we construct are limits in law of a regularized stochastic Euler equation and enjoy an additional $L^2([0,T];H^{-\alpha})$ regularity. For every $p>3/2$ and for certain regularity indices $\alpha \in (0,1/2)$ of the Kraichnan noise, we show also pathwise uniqueness for every $L^p$ initial vorticity. This result is not known without noise.