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

Fan Xu; Lei Zhang; Bin Liu

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arXiv · 2410.17059

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

Fan Xu·Lei Zhang·Bin Liu

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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.

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Fan Xu, Lei Zhang, Bin Liu. 2024-10-22. Global strong solution for the stochastic tamed Chemotaxis-Navier-Stokes system in $\mathbb{R}^3$. https://arxiv.org/abs/2410.17059

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