Boundedness in a 2D chemotaxis-Stokes system with general sensitivity and nonlinear diffusion

This paper deals with the following chemotaxis-Stokes system nt+u⋅∇n=Δnm−∇⋅(nS(x,n,c)⋅∇c),x∈Ω,t>0,ct+u⋅∇c=Δc−nf(c),x∈Ω,t>0,ut=Δu+∇P+n∇ϕ,x∈Ω,t>0,∇⋅u=0,x∈Ω,t>0in a bounded domain Ω⊂R2 with smooth boundary, ϕ∈W1,∞(Ω), f and S are given sufficiently smooth functions with values in [0,∞) and R2×2, respectively. Here S satisfies |S(x,n,c)|1+α and α≥0, the corresponding system possesses a global bounded weak solution for any sufficiently regular initial data (n0,c0,u0) satisfying n0≥0 and c0≥0.