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

Tobias Black,Chunyan Wu

doi:10.1007/s00033-021-01565-z

Abstract

We study a chemotaxis-Stokes system with signal consumption and logistic source terms of the form where kappa ge 0, 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 Nin {2,3} is a prescribed time-independent nonnegative function c_*in C^{2}!left( {{,mathrm{overline{Omega }},}}right) . 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.

Highlights

Chemotaxis, the oriented movement of bacteria and cells in response to a chemical substance in their surrounding environment, is an important motility scheme in nature
An interesting facet of colonies of such chemotactically active bacteria and cells consists of the possibility to spontaneously generate spatial patterns, as witnessed by the experimental findings on the aerobic Bacillus subtilis [6,10,23] and in settings where the attracting signal is produced by the cells themselves [14,45]
Wu condition for the bacteria, a large part of the literature on chemotaxis-fluid systems only considers no-flux conditions for both n and c and a no-slip condition for u

Summary

Introduction

Chemotaxis, the oriented movement of bacteria and cells in response to a chemical substance in their surrounding environment, is an important motility scheme in nature. ZAMP condition for the bacteria ( they even propose mixed boundary conditions distinguishing between the bottom layer of the drop and the fluid–air interface), a large part of the literature on chemotaxis-fluid systems only considers no-flux conditions for both n and c and a no-slip condition for u In this setting, the global solvability of (1.1) is well studied and most of the remaining problems remain in the case of N = 3. Under consideration of different boundary conditions, the knowledge of (1.1) is quite enigmatic, with most of the current results on existence theory only discussing the two-dimensional setting or relying on the inclusion of small changes to (1.1), like logistic growth terms, an enhanced diffusion rate for the bacteria or the consideration of Stokes fluid (i.e., dropping (u · ∇)u in the third equation) and even solutions can often only be obtained with quite mild regularity.

Definition of global weak solutions

Global existence of approximate solutions and essential regularity estimates

Refined a priori information on nε

Regularity estimates for the time derivatives