Global strong solution for the stochastic tamed Chemotaxis-Navier-Stokes system in $\mathbb{R}^3$

TL;DR

This paper proves the existence and uniqueness of global strong solutions for a complex 3D stochastic chemotaxis-Navier-Stokes system with large initial data, using advanced approximation and energy methods.

Contribution

It introduces a novel triple approximation scheme and a stochastic entropy-energy inequality to establish global solutions for the 3D STCNS system.

Findings

Existence of global strong solutions for the 3D STCNS system.

Uniqueness of solutions in both probabilistic and PDE senses.

Development of a stochastic entropy-energy inequality.

Abstract

In this work, we consider the 3D Cauchy problem for a coupled system arising in biomathematics, consisting of a chemotaxis model with a cubic logistic source and the stochastic tamed Navier-Stokes equations (STCNS, for short). Our main goal is to establish the existence and uniqueness of a global strong solution (strong in both the probabilistic and PDE senses) for the 3D STCNS system with large initial data. To achieve this, we first introduce a triple approximation scheme by using the Friedrichs mollifier, frequency truncation operators, and cut-off functions. This scheme enables the construction of sufficiently smooth approximate solutions and facilitates the effective application of the entropy-energy method. Then, based on a newly derived stochastic version of the entropy-energy inequality, we further establish some a priori higher-order energy estimates, which together with the stochastic compactness method, allow us to construct the strong solution for the STCNS system.