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

TL;DR

This paper proves the existence of global very weak solutions for a chemotaxis-fluid system with nonlinear diffusion in three dimensions, under specific conditions on the diffusion exponent, using energy estimates and compactness arguments.

Contribution

It establishes the existence of global very weak solutions for the chemotaxis-fluid system with nonlinear diffusion for the first time in three dimensions, under optimal conditions on the diffusion exponent.

Findings

Global very weak solutions exist for m > 4/3.

Stronger regularity and weak solutions are obtained for m > 5/3.

Energy estimates facilitate compactness and passage to the limit.

Abstract

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.