Well-posedness for a stochastic 2D Euler equation with transport noise

TL;DR

This paper proves the existence and uniqueness of global solutions for a stochastic 2D Euler equation with transport noise, demonstrating the preservation of initial smoothness using viscous approximations and tightness criteria.

Contribution

It introduces a novel approach to establish well-posedness for stochastic Euler equations with transport noise through viscous approximation and compactness arguments.

Findings

Existence of a unique global strong solution

Preservation of initial smoothness of solutions

Use of viscous approximations and tightness criteria

Abstract

We prove the existence of a unique global strong solution for a stochastic two-dimensional Euler vorticity equation for incompressible flows with noise of transport type. In particular, we show that the initial smoothness of the solution is preserved. The arguments are based on approximating the solution of the Euler equation with a family of viscous solutions which is proved to be relatively compact using a tightness criterion by Kurtz.