The small-convection limit in a two-dimensional chemotaxis-Navier–Stokes system

Abstract

This paper deals with an initial-boundary value problem for the chemotaxis-(Navier–)Stokes system

$$\begin{aligned} \left\{ \begin{array}{lcll} n_t + u\cdot \nabla n &{}=&{} \Delta n - \nabla \cdot (n\nabla c), \qquad &{}\quad x\in \Omega , \ t>0, \\ c_t + u\cdot \nabla c &{}=&{}\Delta c - nc, \qquad &{}\quad x\in \Omega , \ t>0, \\ u_t + \kappa (u\cdot \nabla )u &{}=&{} \Delta u - \nabla P + n\nabla \phi , \qquad &{}\quad x\in \Omega , \ t>0, \\ \nabla \cdot u =0, &{} &{} \qquad &{}\quad x\in \Omega , t>0, \end{array} \right. \end{aligned}$$

in a bounded convex domain $\Omega \subset \mathbb {R}^2$ with smooth boundary, with $\kappa \in \mathbb {R}$ and a given smooth potential $\phi :\Omega \rightarrow \mathbb {R}$. It is known that for each $\kappa \in \mathbb {R}$ and all sufficiently smooth initial data this problem possesses a unique global classical solution $(n^{(\kappa )},c^{(\kappa )},u^{(\kappa }))$. The present work asserts that these solutions stabilize to $(n^{(0)},c^{(0)},u^{(0)})$ uniformly with respect to the time variable. More precisely, it is shown that there exist $\mu >0$ and $C>0$ such that whenever $\kappa \in (-1,1)$,

$$\begin{aligned}&\Big \Vert n^{(\kappa )}(\cdot ,t)-n^{(0)}(\cdot ,t)\Big \Vert _{L^\infty (\Omega )} + \Big \Vert c^{(\kappa )}(\cdot ,t)-c^{(0)}(\cdot ,t)\Big \Vert _{L^\infty (\Omega )} \\&\quad +\, \Big \Vert u^{(\kappa )}(\cdot ,t)-u^{(0)}(\cdot ,t)\Big \Vert _{L^\infty (\Omega )} \le C |\kappa | e^{-\mu t} \end{aligned}$$

for all $t>0$. This result thereby provides an example for a rigorous quantification of stability properties in the Stokes limit process, as frequently considered in the literature on chemotaxis-fluid systems in application contexts involving low Reynolds numbers.