Global solvability of chemotaxis-fluid systems with nonlinear diffusion and matrix-valued sensitivities in three dimensions

TL;DR

This paper proves the existence of global solutions for a three-dimensional chemotaxis-fluid system with nonlinear diffusion and matrix-valued sensitivities, extending previous results and identifying optimal conditions for solvability.

Contribution

It extends existing chemotaxis-fluid models to include matrix-valued sensitivities and nonlinear diffusion, establishing conditions for global solvability in three dimensions.

Findings

Existence of at least one global very weak solution under certain conditions.

Existence of at least one global weak solution with stronger exponent conditions.

Optimality of the conditions compared to fluid-free systems.

Abstract

In this work we extend a recent result to chemotaxis fluid systems which include matrix-valued sensitivity functions $S(x,n,c):\Omega\times[0,\infty)^2\to\mathbb{R}^{3\times3}$ in addition to the porous medium type diffusion, which were discussed in the previous work. Namely, we will consider the system \begin{align*} \left\{ \begin{array}{r@{\,}c@{\,}c@{\ }l@{\quad}l@{\quad}l@{\,}c} n_{t}&+&u\cdot\!\nabla n&=\Delta n^m-\nabla\!\cdot(nS(x,n,c)\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. Assuming that $m\geq1$, $\alpha\geq0$ satisfy $m+\alpha>\frac43$, that the matrix-valued function $S(x,n,c):\Omega\times[0,\infty)^2\to\mathbb{R}^{3\times3}$ satisfies $|S(x,n,c)|\leq\frac{S_0}{(1+n)^{\alpha}}$ for some $S_0>0$ and suitably regular nonnegative initial data, we show that the corresponding no-flux-Dirichlet boundary value problem emits at least one global very weak solution. Upon comparison with results for the fluid-free system this condition appears to be optimal. Moreover, imposing a stronger condition for the exponents $m$ and $\alpha$, i.e. $m+2\alpha>\frac{5}{3}$, we will establish the existence of at least one global weak solution in the standard sense.