On Weak-Strong Uniqueness for Stochastic Equations of Incompressible Fluid Flow

Abstract

We introduce a 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 probabilistic sense i.e., the underlying probability space and the driving Wiener process are intrinsic parts of the solution. We first exhibit the relative energy inequality for the incompressible Euler equations driven by a multiplicative noise and then demonstrate the pathwise weak-strong uniqueness principle. Finally, we also provide a sufficient condition, á la Prodi (Ann Mat Pura Appl 48:173–182, 1959) and Serrin (in: Nonlinear problems, University of Wisconsin Press, Madison, Wisconsin, pp 69–98, 1963), for the uniqueness of weak martingale solutions to the stochastic Navier–Stokes system in the class of finite energy weak martingale solutions.