Convergence of a spectral method for the stochastic incompressible Euler equations

Abstract

We propose a spectral viscosity method (SVM) to approximate the incompressible Euler equations driven by amultiplicativenoise. We show that the SVM solution converges to adissipative measure-valued martingalesolution of the underlying problem. These solutions are weak in the probabilistic sensei.e.the probability space and the driving Wiener process are an integral part of the solution. We also exhibit a weak (measure-valued)-strong uniqueness principle. Moreover, we establishstrongconvergence of approximate solutions to the regular solution of the limit system at least on the lifespan of the latter, thanks to the weak (measure-valued)–strong uniqueness principle for the underlying system.