Global classical solutions in chemotaxis(-Navier)-Stokes system with rotational flux term

TL;DR

This paper proves the existence of global classical solutions for a chemotaxis-Navier-Stokes system with rotational flux, showing that small initial chemical concentrations lead to bounded, stabilizing solutions in 2D and 3D domains.

Contribution

It establishes the first global classical solutions for the chemotaxis-Navier-Stokes system with rotational flux under mild conditions, extending previous scalar sensitivity results.

Findings

Global classical solutions exist under small initial chemical concentration.

Solutions remain bounded and converge to equilibrium over time.

The rotational flux case lacks a natural gradient structure, requiring novel estimates.

Abstract

The coupled chemotaxis fluid system \begin{equation} \left\{ \begin{array}{llc} \displaystyle n_t=\Delta n-\nabla\cdot(nS(x,n,c)\cdot\nabla c)-u\cdot\nabla n, &(x,t)\in \Omega\times (0,T),\\ c_t=\Delta c-nc-u\cdot\nabla c , &(x,t)\in\Omega\times (0,T),\\ u_t=\Delta u-\kappa(u\cdot\nabla)u+\nabla P+n\nabla\phi , &(x,t)\in\Omega\times (0,T),\\ \nabla\cdot u=0,&(x,t)\in\Omega\times (0,T), \end{array} \right.(\star) \end{equation} is considered under the no-flux boundary conditions for $n,c$ and the Dirichlet boundary condition for $u$ on a bounded smooth domain $\Omega\subset\mathbb{R}^N$ ($N=2,3$), $\kappa=0,1$. We assume that $S(x,n,c)$ is a matrix-valued sensitivity under a mild assumption such that $|S(x,n,c)|<S_0(c_0)$ with some non-decreasing function $S_0\in C^2((0,\infty))$. It contrasts the related scalar sensitivity case that $(\star)$ does not possess the natural {\em gradient-like} functional structure. Associated estimates based on the natural functional seem no longer available. In the present work, a global classical solution is constructed under a smallness assumption on $\|c_0\|_{L^\infty(\Omega)}$ and moreover we obtain boundedness and large time convergence for the solution, meaning that small initial concentration of chemical forces stabilization.