Global martingale weak solutions for the three-dimensional stochastic chemotaxis-Navier-Stokes system with L\'{e}vy processes

TL;DR

This paper establishes the existence of global martingale solutions for the three-dimensional stochastic chemotaxis-Navier-Stokes system driven by Lévy processes, extending the understanding of such complex stochastic PDEs beyond two dimensions.

Contribution

It introduces a new approach to prove global solutions for the 3D stochastic chemotaxis-Navier-Stokes system with Lévy noise, using entropy-energy inequalities and stochastic compactness methods.

Findings

Proves existence of at least one global martingale solution in 3D.

Develops a stochastic entropy-energy inequality for the system.

Constructs global solutions via regularization and compactness techniques.

Abstract

This paper studies the three-dimensional stochastic chemotaxis-Navier-Stokes (SCNS) system subjected to a L\'{e}vy-type random external force in bounded domain. Up to now, the existing results concerning global solvability of SCNS system mainly concentrated on the case of two spacial dimensions, little is known for the SCNS system in dimension three. We prove in present work that the three-dimensional SCNS system possesses at least one global martingale solution under proper assumptions, which is weak both in the analytical sense and in the stochastic sense. A new stochastic analogue of entropy-energy inequality and an uniform boundedness estimate are derived, which enable us to construct global-in-time approximate solutions from a properly regularized SCNS system via the Contraction Mapping Principle. The proof of the existence of martingale solution is based on the stochastic compactness method and an elaborate identification of the limits procedure, where the Jakubowski-Skorokhod Theorem is applied to deal with the phase spaces equipped with weak topology.