Stochastic Navier-Stokes Equations Driven by Levy noise in unbounded 2D and 3D domains

TL;DR

This paper establishes the existence of martingale solutions for stochastic Navier-Stokes equations driven by Levy noise in unbounded 2D and 3D domains, using approximation and compactness methods.

Contribution

It introduces new existence results for stochastic Navier-Stokes equations with Levy noise in unbounded domains, employing advanced probabilistic and functional analysis techniques.

Findings

Proved existence of martingale solutions in 2D and 3D unbounded domains.

Developed compactness and tightness criteria in specialized function spaces.

Applied a version of the Skorokhod Embedding Theorem for nonmetric spaces.

Abstract

Martingale solutions of stochastic Navier-Stokes equations in 2D and 3D possibly unbounded domains, driven by the L\'evy noise consisting of the compensated time homogeneous Poisson random measure and the Wiener process are considered. Using the classical Faedo-Galerkin approximation and the compactness method we prove existence of a martingale solution. We prove also the compactness and tighness criteria in a certain space contained in some spaces of c\`adl\`ag functions, weakly c\`adl\`ag functions and some Fr\'echet spaces. Moreover, we use a version of the Skorokhod Embedding Theorem for nonmetric spaces.