Existence for stochastic 2D Euler equations with positive $H^{-1}$ vorticity

TL;DR

This paper proves the existence of non-negative vorticity solutions for stochastic 2D Euler equations with transport noise, extending Delort's classical result to the stochastic setting for initial vortex sheets.

Contribution

It extends Delort's deterministic existence result to stochastic 2D Euler equations with transport noise and non-negative vortex sheet initial data.

Findings

Existence of measure- and $H^{-1}$-valued solutions established.

Solutions can start from any non-negative vortex sheet.

Extension of classical results to stochastic case.

Abstract

We prove the existence of non-negative measure- and $H^{-1}$-valued vorticity solutions to the stochastic 2D Euler equations with transport vorticity noise, starting from any non-negative vortex sheet. This extends the result by Delort (J.Amer.Math.Soc. 1991) to the stochastic case.