Prescribed signal concentration on the boundary: Weak solvability in a chemotaxis-Stokes system with proliferation

TL;DR

This paper proves the existence of at least one global weak solution for a chemotaxis-Stokes system with boundary-prescribed signal concentration, incorporating logistic growth and consumption, in two and three-dimensional domains.

Contribution

It introduces a novel boundary condition for the signal concentration in chemotaxis-fluid models and establishes weak solvability results under these conditions.

Findings

Existence of global weak solutions for the system.

Boundary-prescribed signal concentration affects system behavior.

Quadratic decay term ensures boundedness and solvability.

Abstract

We study a chemotaxis-Stokes system with signal consumption and logistic source terms of the form \noindent \begin{align*} \left\{ \begin{array}{r@{\ }l@{\quad}l@{\quad}l@{\,}c} n_{t}+u\cdot\!\nabla n&=\Delta n-\nabla\!\cdot(n\nabla c)+\kappa n-\mu n^{2},\ &x\in\Omega,& t>0,\\ c_{t}+u\cdot\!\nabla c&=\Delta c-nc,\ &x\in\Omega,& t>0,\\ u_{t}&=\Delta u+\nabla P+n\nabla\phi,\ &x\in\Omega,& t>0,\\ \nabla\cdot u&=0,\ &x\in\Omega,& t>0,\\ \big(\nabla n-n\nabla c\big)\cdot\nu&=0,\quad c=c_{\star}(x),\quad u=0, &x\in\partial\Omega,& t>0, \end{array}\right. \end{align*} where $\kappa\geq0$, $\mu>0$ and, in contrast to the commonly investigated variants of chemotaxis-fluid systems, the signal concentration on the boundary of the domain $\Omega\subset\mathbb{R}^N$ with $N\in\{2,3\}$, is a prescribed time-independent nonnegative function $c_{\star}\in C^{2}\!\big(\overline{\Omega}\big)$. Making use of the boundedness information entailed by the quadratic decay term of the first equation, we will show that the system above has at least one global weak solution for any suitably regular triplet of initial data.