Global existence and convergence rates to a chemotaxis-fluids system with mixed boundary conditions

Abstract

In this paper, we investigate the large time behavior of strong solutions to a chemotaxis-fluids system in an unbounded domain with mixed boundary conditions. Based on the anisotropic Lp technique, the elliptic estimates and Stokes estimates, we first establish the global existence of strong solution around the equilibrium state (0,csatn,0) with the help of the continuity arguments, where csatn is the saturation value of oxygen inside the fluid. Then we use De Giorgi's technique and energy method to show that such a solution will converge to (0,csatn,0) with an explicit convergence rate in the chemotaxis-free case. Our assumptions and results are consistent with the experimental descriptions and the numerical analysis. The novelty here consists of deriving some new elliptic estimates and Stokes estimates, and choosing a suitable weight in De Giorgi's technique to deal with the mixed boundary conditions.