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&=Δn-\nabla\!\cdot(\frac{n}{c}\nabla c),\ &x\inΩ,& t>0, c_{t}&+&u\cdot\!\nabla c&=Δc-nc,\ &x\inΩ,& t>0, u_{t}&+&\nabla P&=Δu+n\nablaϕ,\ &x\inΩ,& t>0, &&\nabla\cdot u&=0,\ &x\inΩ,& 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 $\varphi \in C^2\left(\bar{\Omega }\right)$. 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 $L^2\left(\Omega \right)$.

35 pages

Cited by in corpus (4)

• On the global generalized solvability of a chemotaxis model with signal absorption and logistic growth terms
• Chemotaxis(-fluid) systems with logarithmic sensitivity and slow consumption: global generalized solutions and eventual smoothness
• Uniform $L^p$ estimates for solutions to the inhomogeneous 2D Navier-Stokes equations and application to a chemotaxis-fluid system with local sensing
• On a Repulsion Keller--Segel System with a Logarithmic Sensitivity