Eventual smoothness of generalized solutions to a singular chemotaxis-Stokes system

Abstract

We study the chemotaxis-fluid system \begin{align*} \left\{ \begin{array}{r@{\,}c@{\,}c@{\ }l@{\quad}l@{\quad}l@{\,}c} n_{t}&+&u\cdot\!\nabla n&=\Delta n-\nabla\!\cdot(\frac{n}{c}\nabla c),\ &x\in\Omega,& t>0, c_{t}&+&u\cdot\!\nabla c&=\Delta c-nc,\ &x\in\Omega,& t>0, u_{t}&+&\nabla P&=\Delta u+n\nabla\phi,\ &x\in\Omega,& t>0, &&\nabla\cdot u&=0,\ &x\in\Omega,& t>0, \end{array}\right. \end{align*} under homogeneous Neumann boundary conditions for $n$ and $c$ and homogeneous Dirichlet boundary conditions for $u$, where $\Omega\subset\mathbb{R}^2$ is a bounded domain with smooth boundary and $\phi\in C^{2}\left(\bar{\Omega}\right)$. From recent results it is known that for suitable regular initial data, the corresponding initial-boundary value problem possesses a global generalized solution. We will show that for small initial mass $\int_{\Omega}\!n_0$ these generalized solutions will eventually become classical solutions of the system and obey certain asymptotic properties. Moreover, from the analysis of certain energy-type inequalities arising during the investigation of the eventual regularity, we will also derive a result on global existence of classical solutions under assumption of certain smallness conditions on the size of $n_0$ in $L^1\!\left(\Omega\right)$ and in $L\log L\!\left(\Omega\right)$, $u_0$ in $L^4\!\left(\Omega\right)$, and of $\nabla c_0$ in $L^2\!\left(\Omega\right)$.