Global boundedness of weak solutions for a three-dimensional chemotaxis-Stokes system with nonlinear diffusion and rotation

Abstract

This paper deals with the chemotaxis-Stokes system with nonlinear diffusion and rotation: nt+u⋅∇n=Δnm−∇⋅(nS(x,n,c)⋅∇c), ct+u⋅∇c=Δc−nc, ut+∇P=Δu+n∇ϕ+f(x,t) and ∇⋅u=0, in a bounded domain Ω⊂R3, where m>0, and ϕ:Ω¯→R, f:Ω¯×[0,∞)→R3 and S:Ω¯×[0,∞)2→R3×3 are given sufficiently smooth functions such that f is bounded in Ω×(0,∞) and S satisfies |S(x,n,c)|≤S0(c)(1+n)−α for all (x,n,c)∈Ω¯×[0,∞)2 with α>0 and some nondecreasing function S0:[0,∞)→[0,∞). It is shown that if m+α>109 and m+54α>98, then for any reasonably smooth initial data, the corresponding Neumann-Neumann-Dirichlet initial-boundary problem possesses a globally bounded weak solution. This extends the previous global boundedness result for m>98 and α=0[43], and improves that for m≥1 and m+α>76[34], or for m+α>76 in the associated fluid-free system [31]. Our proof consists at its core in using, inter alia, the maximal Sobolev regularity theory to elaborately derive some spatio-temporal estimates for the signal and the fluid equations so as to decouple the system.