Boundedness in a three-dimensional chemotaxis-Stokes system with tensor-valued sensitivity

This paper is concerned with the chemotaxis-Stokes system equation(⋆⋆){nt+u⋅∇n=Δn−∇⋅(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 under no-flux boundary conditions in a bounded domain Ω⊂R3Ω⊂R3 with smooth boundary. Here SS is a matrix-valued sensitivity satisfying |S(x,n,c)|0 and α>0α>0. We shall first establish the boundedness of (⋆⋆) through a series of a priori estimate, and then propose the stabilization of the classical solution. This result gives a definite answer to the conjecture in Remark 1.2 by Y. Wang and X. Cao (2015) and extends the main result of that paper.