Convergence of a spectral method for the stochastic incompressible Euler equations

TL;DR

This paper introduces a spectral viscosity method for approximating stochastic incompressible Euler equations, demonstrating convergence to measure-valued solutions and establishing a weak-strong uniqueness principle.

Contribution

It develops a novel spectral viscosity method for stochastic Euler equations and proves convergence to measure-valued solutions with a weak-strong uniqueness result.

Findings

Convergence of the spectral viscosity method to measure-valued solutions.

Establishment of a weak (measure-valued)-strong uniqueness principle.

Strong convergence of approximate solutions to regular solutions during their lifespan.

Abstract

We propose a spectral viscosity method (SVM) to approximate the incompressible Euler equations driven by a multiplicative noise. We show that SVM solution converges to a dissipative measure-valued martingale solution. These solutions are weak in the probabilistic sense i.e. the probability space and the driving Wiener process are an integral part of the solution. We also exhibit weak (measure-valued)-strong uniqueness principle. Moreover, we establish strong convergence 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.