Global Very Weak Solutions to a Chemotaxis-Fluid System with Nonlinear Diffusion

Abstract

We consider the chemotaxis-fluid system given by $n_{t}+u\cdot\!\nabla n=\Delta n^m-\nabla\!\cdot(n\nabla c)$, $c_{t}+u\cdot\!\nabla c=\Delta c-c+n$, $u_{t}+(u\cdot\nabla)u=\Delta u+\nabla P+n\nabla\phi$, and $\nabla\cdot u=0$, for $x\in\Omega$ and $t>0$, where $\Omega\subset\mathbb{R}^3$ is a bounded domain 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 this system 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 the system above the condition appears to be optimal with respect to global existence. In the 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 the system above. 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.