Global bounded weak solutions for a chemotaxis-Stokes system with nonlinear diffusion and rotation

Abstract

In this paper, we discuss the following chemotaxis-Stokes system with nonlinear diffusion and rotation{nt+u⋅∇n=Δnm−∇⋅(nS(x,n,c)⋅∇c),x∈Ω,t>0,ct+u⋅∇c=Δc−nc,x∈Ω,t>0,ut+∇P=Δu+n∇ϕ,x∈Ω,t>0,∇⋅u=0,x∈Ω,t>0 with m>0, where Ω is a bounded domain in R3, S(x,n,c) is a chemotactic sensitivity tensor satisfying S∈C2(Ω¯×[0,∞)2;R3×3) and |S(x,n,c)|≤(1+n)−αS0(c) for all (x,n,c)∈Ω×[0,∞)2 with α≥0 and some nondecreasing function S0:[0,∞)→[0,∞). We can show that if m+α>109, then for any reasonably smooth initial data, the corresponding initial-boundary problem possesses a globally bounded weak solution. The result extends the previous global boundedness result for m+α>109, α>0 and m+54α>98[15], and m>98 and α=0[24]. Without using the conventional free-energy inequality see [15,24], we use a new iterative bootstrap procedure to estimate the bounds on ∫Ωnεp for p>32, which is a crucial step to obtain our main result.