Weak solutions of Navier-Stokes equation with purely discontinuous Lévy noise

In this paper we prove the existence of weak martingale solutions to the stochastic Navier-Stokes Equations driven by pure jump Lévy 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 authors in their joint paper with E. Hausenblas, see [11].