Well-posedness and large deviations for 2-D Stochastic Navier-Stokes equations with jumps

TL;DR

This paper establishes the existence, uniqueness, and large deviation principles for 2D stochastic Navier-Stokes equations driven by Lévy noise, advancing the understanding of their probabilistic behavior and control properties.

Contribution

It proves well-posedness, develops a Girsanov theorem for jump noise, and derives large deviation principles for 2D SNSEs with Lévy noise, which are novel contributions.

Findings

Existence and uniqueness of solutions under Lévy noise

Girsanov theorem for Poisson random measures

Large deviation principles for SNSE solutions

Abstract

The aim of this paper is threefold. Firstly, we prove the existence and the uniqueness of a global strong (in both the probabilistic and the PDE senses) $\mathrm{H}^{1}_2$-valued solution to the 2D stochastic Navier-Stokes equations (SNSEs) driven by a multiplicative L\'evy noise under the natural Lipschitz on balls and linear growth assumptions on the jump coefficient. Secondly, we prove a Girsanov-type theorem for Poisson random measures and apply this result to a study of the well-posedness of the corresponding stochastic controlled problem for these SNSEs. Thirdly, we apply these results to establish a Freidlin-Wentzell-type large deviation principle for the solutions of these SNSEs by employing the weak convergence method introduced in papers [16][18].