We consider the chemotaxis-fluid system \begin{ align}\label{ star}\tag{ $\diamondsuit$} \left\{ \begin{ array}{ r@{ \,}c@{ \,}c@{ \ }l@{ \quad}l@{ \quad}l@{ \,}c} n_{ t}&+&u\cdot\!\nabla n&=\Delta n^m-\nabla\!\cdot(n\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 $\Omega\subset\mathbb{ R}^3$ with smooth boundary and $m>1$. Assuming $m>\frac{ 4}{ 3}$ and sufficiently regular nonnegative initial data, we ensure the existence of global solutions to the no-flux-Dirichlet boundary value problem for \eqref{ star} under a suitable notion of very weak solvability, which in different variations has been utilized in the literature before. Comparing this with known results for the fluid-free setting of \eqref{ star} the condition appears to be optimal with respect to global existence. In case of the stronger assumption $m>\frac{ 5}{ 3}$ we moreover establish the existence of at least one global weak solution in the standard sense. In our analysis we investigate a functional of the form $\int_{ \Omega}\! n^{ m-1}+\int_{ \Omega}\! c^2$ to obtain a spatio-temporal $L^2$ estimate on $\nabla n^{ m-1}$, which will be the starting point in deriving a series of compactness properties for a suitably regularized version of \eqref{ star}. As the regularity information obtainable from these compactness results vary depending on the size of $m$, we will find that taking $m>\frac{ 5}{ 3}$ will yield sufficient regularity to pass to the limit in the integrals appearing in the weak formulation, while for $m>\frac{ 4}{ 3}$ we have to rely on milder regularity requirements making only very weak solutions attainable.
Categories: math.AP