A chemotaxis-fluid model driven by L\'{e}vy noise in $\mathbb{R}^2$

TL;DR

This paper establishes the existence and uniqueness of global solutions for a coupled chemotaxis-Navier-Stokes system influenced by Lévy noise in two-dimensional space, advancing understanding of stochastic biological-fluid models.

Contribution

It introduces a novel stochastic Lyapunov functional approach to prove global solutions for chemotaxis-fluid systems with Lévy noise, including pathwise uniqueness.

Findings

Existence of global martingale solutions under Lévy noise.

Pathwise uniqueness of strong solutions with specific noise assumptions.

Development of stochastic Lyapunov functional inequality techniques.

Abstract

In this paper, we investigate the existence and uniqueness of global solutions to the Cauchy problem for a coupled stochastic chemotaxis-Navier-Stokes system with multiplicative L\'{e}vy noises in $\mathbb{R}^2$. The existence of global martingale solutions is proved under a framework that is based on the Faedo-Galerkin approximation scheme and stochastic compactness method, where the verification of tightness depends crucially on a novel stochastic version of Lyapunov functional inequality and proper compactness criteria in Fr\'{e}chet spaces. A pathwise uniqueness result is also established with suitable assumption on the jump noises, which indicates that the considered system admits a unique global strong solution.