Weak solutions of Navier-Stokes Equation with purely discontinuous L\'evy Noise

TL;DR

This paper establishes the existence of weak martingale solutions for the stochastic Navier-Stokes equations driven by purely discontinuous Lévy noise, using a novel approach involving Galerkin approximations and martingale representation.

Contribution

It introduces a new method for proving solutions to stochastic PDEs driven by jump processes, differing from previous approaches.

Findings

Proves existence of weak martingale solutions for SNSE with Lévy noise.

Develops a new approach using Galerkin approximations and martingale representation.

Provides a framework for analyzing stochastic PDEs with discontinuous noise.

Abstract

In this paper we prove the existence of weak martingale solutions to the stochastic Navier-Stokes Equations driven by pure jump L\'evy processes. Our proof consists of two parts. In the first one, mostly classical, we recall a priori estimates, from the paper by the third named author, for solutions to suitable constructed Galerkin approximations and we use the Jakubowski-Skorokhod Theorem to find a sequence of processes on a new probability space convergent point-wise to a limit process. In the second one, we show that the limit process is a weak martingale solution to the SNSEs by using an approach of Kallianpur and Xiong. The core of this method consists of a proof that a certain natural process on the new probability space is a purely discontinuous martingale and then to use a suitable representation theorem. In this way we propose a method of proving solutions to stochastic PDEs which is different from the method used recently by the first and fourth named author in their joint paper with E.\ Hausenblas, see \cite{Brz+Haus+Raza_2018_reaction_diffusion}.