Asymptotic dynamics on a chemotaxis-Navier–Stokes system with nonlinear diffusion and inhomogeneous boundary conditions

Abstract

The diffusion of cells in a viscous incompressible fluid (e.g. water) may be viewed like movement in a porous medium and there is a bidirectorial oxygen exchange between water and their surrounding air in thin fluid layers near the air–water contact surface. This leads to the following chemotaxis-Navier–Stokes system with nonlinear diffusion: [Formula: see text] endowed with the inhomogeneous boundary conditions [Formula: see text] and the initial data [Formula: see text] in [Formula: see text], where the incoming oxygen [Formula: see text] is non-negative, and the outgoing oxygen molecule is modeled by [Formula: see text] with positive coefficient [Formula: see text]. In this paper, we investigate the asymptotic dynamics of the above system in a bounded domain [Formula: see text] with the smooth boundary [Formula: see text]. We will show that arbitrary porous medium diffusion mechanism [Formula: see text] can inhibit the singularity formation. In the incoming oxygen-free case, we further prove that the solution will stabilize to the unique mass-preserving spatial equilibrium [Formula: see text] in the sense that as [Formula: see text], [Formula: see text] hold uniformly with respect to [Formula: see text], where [Formula: see text].