Global classical small-data solutions for a three-dimensional chemotaxis Navier-Stokes system involving matrix-valued sensitivities

TL;DR

This paper proves the existence of global classical solutions for a three-dimensional chemotaxis-Navier-Stokes system with matrix-valued sensitivities under small initial data conditions, and analyzes their decay properties.

Contribution

It establishes the first global classical solutions for this chemotaxis-fluid system with matrix sensitivities in 3D, extending previous results to more complex sensitivities.

Findings

Global classical solutions exist under small initial data.

Solutions exhibit decay over time.

Results apply to both 2D and 3D cases.

Abstract

The coupled chemotaxis fluid system \begin{equation} \left\{ \begin{array}{llc} n_t=\Delta n-\nabla\cdot(n S(x,n,c)\cdot\nabla c)-u\cdot\nabla n, &(x,t)\in \Omega\times (0,T), \displaystyle c_t=\Delta c-nc-u\cdot\nabla c, &(x,t)\in\Omega\times (0,T), \displaystyle u_t=\Delta u-(u\cdot\nabla )u+\nabla P+n\nabla\Phi,\quad \nabla\cdot u=0, &(x,t)\in\Omega\times (0,T), \displaystyle \nabla c\cdot\nu=(\nabla n-nS(x,n,c)\cdot\nabla c)\cdot\nu=0, \;\; u=0,&(x,t)\in \partial\Omega\times (0,T), n(x,0)=n_{0}(x),\quad c(x,0)=c_{0}(x),\quad u(x,0)=u_0(x) & x\in\Omega, \end{array} \right. \end{equation} where $S\in (C^2(\bar{\Omega}\times [0,\infty)^2))^{N\times N}$, is considered in a bounded domain $\Omega\subset\mathbb{R}^N$, $N\in\{2,3\}$, with smooth boundary. We show that it has global classical solutions if the initial data satisfy certain smallness conditions and give decay properties of these solutions.