Dissipative solutions to the stochastic Euler equations

TL;DR

This paper introduces dissipative martingale solutions for the stochastic Euler equations, constructed via vanishing viscosity limits, and establishes a weak-strong uniqueness principle under stochastic forcing.

Contribution

It develops a new framework for solutions to stochastic Euler equations using generalized Young measures and proves weak-strong uniqueness.

Findings

Existence of dissipative martingale solutions via vanishing viscosity

Solutions satisfy an energy inequality

Weak-strong uniqueness holds for these solutions

Abstract

We study the three-dimensional incompressible Euler equations subject to stochastic forcing. We develop a concept of dissipative martingale solutions, where the nonlinear terms are described by generalised Young measures. We construct these solutions as the vanishing viscosity limit of solutions to the corresponding stochastic Navier-Stokes equations. This requires a refined stochastic compactness method incorporating the generalised Young measures. Our solutions satisfy a form of the energy inequality which gives rise to a weak-strong uniqueness result (pathwise and in law). A dissipative martingale solution coincides (pathwise or in law) with the strong solution as soon as the latter exists.