Global existence and stabilization in a degenerate chemotaxis-Stokes system with mildly strong diffusion enhancement

Abstract

A class of chemotaxis-Stokes systems generalizing the prototype{nt+u⋅∇n=∇⋅(nm−1∇n)−∇⋅(n∇c),ct+u⋅∇c=Δc−nc,ut+∇P=Δu+n∇ϕ,∇⋅u=0, is considered in bounded convex three-dimensional domains, where ϕ∈W2,∞(Ω) is given.The paper develops an analytical approach which consists in a combination of energy-based arguments and maximal Sobolev regularity theory, and which allows for the construction of global bounded weak solutions to an associated initial-boundary value problem under the assumption that(0.1)m>98. Moreover, the obtained solutions are shown to approach the spatially homogeneous steady state (1|Ω|∫Ωn0,0,0) in the large time limit.This extends previous results which either relied on different and apparently less significant energy-type structures, or on completely alternative approaches, and thereby exclusively achieved comparable results under hypotheses stronger than (0.1).