Global classical small-data solutions for a three-dimensional chemotaxis   Navier-Stokes system involving matrix-valued sensitivities
  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. 
Xinru Cao and Johannes Lankeit
math.AP