Global Existence and Decay Rates to a Self‐Consistent Chemotaxis‐Fluid System Around Nonconstant Equilibrium

ABSTRACT This paper addresses a self‐consistent chemotaxis‐fluid model that elucidates chemotactic movement within a viscous fluid by taking into account both the chemotactic force acting on cells and the gravitational force exerted on the fluid. Additionally, it simultaneously considers the frictional forces and their reactions between cells and the surrounding moving fluid, which induces more nonlinearity and a stronger coupling mechanism for the system. The novelty of this model is that it posits a nonconstant equilibrium , where is a constant and is the gravitational potential function. Under basic regular assumptions on parametric functions , we first show global‐in‐time existence of low‐regularity solutions around this nonconstant equilibrium. We also obtain global existence and explicit temporal decay rates of high‐regularity solutions near the same inhomogeneous state provided that the external potential gradient is suitably weak. To the best of our knowledge, it seems to be the first work addressing the global well‐posedness and large‐time behavior of solutions to chemotaxis‐fluid interaction models, especially to self‐consistent variants, around nonconstant equilibrium.