Stochastic hydrodynamic-type evolution equations driven by L\'{e}vy noise in 3D unbounded domains - abstract framework and applications

TL;DR

This paper establishes the existence of martingale solutions for stochastic hydrodynamic equations driven by Lévy noise in 3D unbounded domains, using advanced probabilistic and functional analysis techniques, with applications to Navier-Stokes, MHD, and Boussinesq equations.

Contribution

It develops an abstract framework for solving stochastic hydrodynamic equations with Lévy noise in unbounded domains, overcoming compactness issues with Fréchet spaces and non-metric space techniques.

Findings

Proved existence of martingale solutions in 3D unbounded domains.

Applied the framework to stochastic Navier-Stokes, MHD, and Boussinesq equations.

Utilized non-metric space compactness and Skorokhod theorems for solution construction.

Abstract

The existence of martingale solutions of the hydrodynamic-type equations in 3D possibly unbounded domains is proved. The construction of the solution is based on the Faedo-Galerkin approximation. To overcome the difficulty related to the lack of the compactness of Sobolev embeddings in the case of unbounded domain we use certain Fr\'{e}chet space. We use also compactness and tightness criteria in some nonmetrizable spaces and a version of the Skorokhod Theorem in non-metric spaces. The general framework is applied to the stochastic Navier-Stokes, magneto-hydrodynamic (MHD) and the Boussinesq equations.