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.