Probabilistically Strong Solutions to Stochastic Euler Equations

TL;DR

This paper proves the existence and convergence of probabilistically strong solutions for stochastic Euler equations, solving a key open problem and introducing energy-variational solutions for stochastic fluid dynamics.

Contribution

It establishes the existence of probabilistically strong solutions for stochastic Euler equations with energy inequalities, extending previous results to general initial data and transport noise.

Findings

Existence of probabilistically strong solutions for stochastic Euler equations.

Convergence of solutions from Navier--Stokes to Euler equations in the vanishing viscosity limit.

Introduction of energy-variational solutions in stochastic fluid dynamics.

Abstract

In this paper, we establish the existence of probabilistically strong, measure-valued solutions for the stochastic incompressible Navier--Stokes equations and prove their convergence, in the vanishing viscosity limit, to probabilistically strong solutions for the stochastic incompressible Euler equations. In particular, this solves the open problem of constructing probabilistically strong solutions for the stochastic Euler equations that satisfy the energy inequality for general $L^2$ initial data. We introduce the concept of energy-variational solutions in the stochastic context in order to treat the nonlinearities without changing the probability space. Furthermore, we extend these results to fluids driven by transport noise.