| Abstract: |
| We discuss global (very) weak solutions to chemotaxis-fluid systems of the form
\begin{align*}
\left\{
\begin{array}{r@{\,}c@{\,}c@{\ }l@{\quad}l@{\quad}l@{\,}c}
n_{t}&+&u\cdot\!\nabla n&=\nabla\!\cdot\big(D(n)\nabla n-S(x,n,c)\nabla c\big),\ &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 $\Omega\subset\mathbb{R}^3$ with smooth boundary. The prototypical choices involve linear diffusion or porous medium type diffusion, i.e. $D(n)=m n^{m-1}$ with $m\geq 1$ and tensor-valued sensitivities $S\in C^2\big(\overline{\Omega}\times[0,\infty)^2;\mathbb{R}^{3\times 3}\big)$ satisfying $|S(x,n,c)|\leq \frac{C_S}{(n+1)^\alpha}$ with $\alpha\geq0$ and constant $C_S>0$. |
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