On weak-strong uniqueness for stochastic equations of incompressible fluid flow

TL;DR

This paper introduces a new framework for analyzing stochastic incompressible fluid equations, establishing weak-strong uniqueness and conditions for solution uniqueness in stochastic Euler and Navier-Stokes systems.

Contribution

It develops a dissipative measure-valued martingale solution concept and proves a pathwise weak-strong uniqueness principle for stochastic Euler equations.

Findings

Established a relative energy inequality for stochastic Euler equations.

Proved pathwise weak-strong uniqueness for these equations.

Provided a Prodi-Serrin type condition for uniqueness of stochastic Navier-Stokes solutions.

Abstract

We introduce a novel concept of dissipative measure-valued martingale solution to the stochastic Euler equations describing the motion of an inviscid incompressible fluid. These solutions are characterized by a parametrized Young measure and a concentration defect measure in the total energy balance. Moreover, they are weak in the probablistic sense i.e., the underlying probablity space and the driving Wiener process are intrinsic part of the solution. In a significant departure from the existing literature, we first exhibit the relative energy inequality for the incompressible Euler equations driven by a multiplicative noise, and then demonstrate pathwise weak-strong uniqueness principle. Finally, we also provide a sufficient condition, a la Prodi and Serrin, for the uniqueness of weak martingale solutions to stochastic Naiver-Stokes system in the class of finite energy weak martingale solutions.