Global classical solutions of a 3D chemotaxis-Stokes system with rotation

This paper considers the chemotaxis-Stokes system$$\begin{cases}\displaystyle n_t+u\cdot\nabla n=\Deltan-\nabla\cdot(nS(x,n,c)\cdot\nabla c),&(x,t)\in \Omega\times (0,T),\\\displaystylec_t+u\cdot\nabla c=\Delta c-nc, &(x,t)\in\Omega\times (0,T),\qquad(\star)\\\displaystyleu_t=\Delta u+\nabla P+n\nabla\phi , &(x,t)\in\Omega\times (0,T),\\\nabla\cdot u=0,&(x,t)\in\Omega\times (0,T).\end{cases}$$ under no-flux boundary conditions in a boundeddomain $\Omega \subset \mathbb{R}^3$ with smooth boundary. Here $S$ isa matrix-valued sensitivity satisfying$|S(x,n,c)|0$ and$\alpha>0$. Although $(\star)$ does not possess the naturalgradient-like functional structure available when $S$ reduces to ascalar function, we can still establish a new energy typeinequality. Based on this inequality we achieve a coupled estimatefor arbitrarily high Lebesgue norms of $n$ and $\nabla c$. Thishelps us to finally obtain the existence of a global classicalsolution when $\alpha$ is bigger than $\frac{1}{6}$.