Global very weak solutions to a chemotaxis-fluid system with nonlinear diffusion

摘要

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 abounded domain $\Omega\subset\mathbb{R}^3$ with smooth boundary and $m>1$. Wewill introduce a notion of very weak solvability and prove that if$m>\frac{4}{3}$, then for any sufficiently regular nonnegative initial datathere exists at least one global very weak solution of a no-flux-Dirichletboundary value problem for \eqref{star}. Moreover, if $m>\frac{5}{3}$ weestablish the existence of at least one global weak solution in the standardsense. 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 ofcompactness properties for a suitably regularized version of \eqref{star}. Asthe regularity information obtainable from these compactness results varydepending on the size of $m$, we will find that taking $m>\frac{5}{3}$ willyield sufficient regularity to pass to the limit in the integrals appearing inthe weak formulation, while for $m>\frac{4}{3}$ we have to rely on milderregularity requirements making only very weak solutions attainable.