Stabilization in a two-dimensional chemotaxis-Navier–Stokes system

Abstract

This paper deals with an initial-boundary value problem for the system $$\left\{ \begin{array}{llll} n_t + u\cdot\nabla n &=& \Delta n -\nabla \cdot (n\chi(c)\nabla c), \quad\quad & x\in\Omega, \, t > 0,\\ c_t + u\cdot\nabla c &=& \Delta c-nf(c), \quad\quad & x\in\Omega, \, t > 0,\\ u_t + \kappa (u\cdot \nabla) u &=& \Delta u + \nabla P + n \nabla\phi, \qquad & x\in\Omega, \, t > 0,\\ \nabla \cdot u &=& 0, \qquad & x\in\Omega, \, t > 0,\end{array} \right.$$ which has been proposed as a model for the spatio-temporal evolution of populations of swimming aerobic bacteria. It is known that in bounded convex domains $${\Omega \subset \mathbb{R}^2}$$ and under appropriate assumptions on the parameter functions χ, f and ϕ, for each $${\kappa\in\mathbb{R}}$$ and all sufficiently smooth initial data this problem possesses a unique global-in-time classical solution. The present work asserts that this solution stabilizes to the spatially uniform equilibrium $${(\overline{n_0},0,0)}$$ , where $${\overline{n_0}:=\frac{1}{|\Omega|} \int_\Omega n(x,0)\,{\rm d}x}$$ , in the sense that as t→∞, $$n(\cdot,t) \to \overline{n_0}, \qquad c(\cdot,t) \to 0 \qquad \text{and}\qquad u(\cdot,t) \to 0$$ hold with respect to the norm in $${L^\infty(\Omega)}$$ .