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=Δn^m-\nabla\cdot(nS(x,n,c)\cdot\nabla c),\quad x\in Ω, t>0, c_t+u\cdot\nabla c=Δc-nc, x\in Ω, t>0,\\ u_t+\nabla P=Δu+n\nabla ϕ,x\in Ω, t>0,\\ \nabla\cdot u=0, x\in Ω, t>0, \end{array}\right. \end{equation} where $\Omega \subseteq {\mathbb{R}}^3$ is a bounded convex domain with smooth boundary. Here $S\in C^2(\bar{\Omega }\times [0,\infty {)}^2;{\mathbb{R}}^{3\times 3})$ is a matrix with $s_{i,j}\in C^1(\bar{\Omega }\times [0,\infty )\times [0,\infty )).$ Moreover, $|S(x,n,c)|\le S_0(c)$ 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 $$ \|u(\cdot,t)\|_{L^\infty(Ω)} +\|c(\cdot,t)\|_{W^{1,\infty}(Ω)}+\|n(\cdot,t)\|_{L^\infty(Ω)} \leq C ~~\mbox{for all}~~ t\in(0,T) $$ is valid with some $C(T)>0$.

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• Global existence to a $3D$ chemotaxis-Navier-stokes system with nonlinear diffusion and rotation