Global solvability of chemotaxis-fluid systems with nonlinear diffusion and matrix-valued sensitivities in three dimensions

In this work we extend a recent result to chemotaxis fluid systems which include matrix-valued sensitivity functions $S(x,n,c):\mathrm{\Omega }\times [0,\mathrm{\infty }{)}^2\to {\mathbb{R}}^{3\times 3}$ in addition to the porous medium type diffusion, which were discussed in the previous work. Namely, we will consider the system \begin{align*} \left{ \begin{array}{r@{,}c@{,}c@{\ }l@{\quad}l@{\quad}l@{,}c} n_{t}&+&u\cdot!\nabla n&=\Delta n^m-\nabla!\cdot(nS(x,n,c)\nabla c),
&x\in\Omega,& t>0,\ c_{t}&+&u\cdot!\nabla c&=\Delta c-c+n,\ &x\in\Omega,& t>0,\ u_{t}&+&(u\cdot\nabla)u&=\Delta u+\nabla P+n\nabla\phi,\ &x\in\Omega,& t>0,\ &&\nabla\cdot u&=0,\ &x\in\Omega,& t>0, \end{array}\right. \end{align*} in a bounded domain $\mathrm{\Omega }\subset {\mathbb{R}}^3$ with smooth boundary. Assuming that $m\ge 1$, $\alpha \ge 0$ satisfy $m+\alpha >\frac{4}{3}$, that the matrix-valued function $S(x,n,c):\mathrm{\Omega }\times [0,\mathrm{\infty }{)}^2\to {\mathbb{R}}^{3\times 3}$ satisfies $\mathrm{\mid }S(x,n,c)\mathrm{\mid }\le \frac{S_0}{(1+n{)}^{\alpha }}$ for some $S_0>0$ and suitably regular nonnegative initial data, we show that the corresponding no-flux-Dirichlet boundary value problem emits at least one global very weak solution. Upon comparison with results for the fluid-free system this condition appears to be optimal. Moreover, imposing a stronger condition for the exponents $m$ and $\alpha$, i.e. $m+2\alpha >\frac{5}{3}$, we will establish the existence of at least one global weak solution in the standard sense.