A further study on a 3D chemotaxis–Stokes system with tensor-valued sensitivity

Abstract

We consider the three-dimensional chemotaxis–Stokes system 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>0under no-flux boundary conditions in a bounded domain Ω⊂R3 with smooth boundary. Here the tensor-valued sensitivity S fulfills |S(x,n,c)|≤Cs(1+n)−α with some Cs>0 and α>0.It has been shown in Wang–Pang–Li (2016) that if α>16, this system possesses a global bounded classical solution for any given initial data. We shall improve this result to the case of arbitrary α>0, and thus solve the problem left in Wang–Cao (2015).