A new approach toward locally bounded global solutions to a $3D$ chemotaxis-stokes system with nonlinear diffusion and rotation

Abstract

We consider a degenerate quasilinear chemotaxis--Stokes type involving rotation in the aggregative term, \begin{equation} \left\{ \begin{array}{l} n_t+u\cdot\nabla n=\Delta n^m-\nabla\cdot(nS(x,n,c)\cdot\nabla c),\quad x\in \Omega, t>0, c_t+u\cdot\nabla c=\Delta c-nc, x\in \Omega, t>0,\\ u_t+\nabla P=\Delta u+n\nabla \phi ,x\in \Omega, t>0,\\ \nabla\cdot u=0, x\in \Omega, t>0, \end{array}\right. \end{equation} where $\Omega\subseteq \mathbb{R}^3$ is a bounded convex domain with smooth boundary. Here $ S\in C^2(\bar{\Omega}\times[0,\infty)^2;\mathbb{R}^{3\times3})$ is a matrix with $s_{i,j}\in C^1( \bar{\Omega} \times [0, \infty)\times[0, \infty)).$ Moreover, $|S(x,n,c)| \leq S_0(c)$ for all $(x,n,c)\in \bar{\Omega} \times [0, \infty)\times[0, \infty)$ with $S_0(c)$ nondecreasing on $[0,\infty)$. If $$m>\frac{9}{8}, $$ then for all reasonably regular initial data, a corresponding initial-boundary value problem for $(0.1)$ possesses a globally defined weak solution $(n,c,u)$. Moreover, for any fixed $T > 0$ this solution is bounded in $\Omega\times (0,T)$ in the sense that $$ \|u(\cdot,t)\|_{L^\infty(\Omega)} +\|c(\cdot,t)\|_{W^{1,\infty}(\Omega)}+\|n(\cdot,t)\|_{L^\infty(\Omega)} \leq C ~~\mbox{for all}~~ t\in(0,T) $$ is valid with some $C(T) > 0$.