Global existence and boundedness in a 2D Keller-Segel-Stokes system

In this paper, we investigate the Keller-Segel-Stokes system (K-S-S): {nt+u⋅∇n=Δn−∇⋅(n∇c),x∈Ω,t>0,τct+u⋅∇c=Δc−c+n,x∈Ω,t>0,ut+∇P=Δu+n∇ϕ,x∈Ω,t>0,∇⋅u=0,x∈Ω,t>0, with no-flux boundary conditions for n and c as well as no-slip boundary condition for u in a bounded domain Ω⊂R2 with smooth boundary. For τ=0, it was shown by Lorz (2012) that the corresponding initial value problem has global-in time solutions for initial mass of cells below some specified value, which may be larger than the well known critical mass of the corresponding fluid-free system. For the case τ=1, i.e, the parabolic-parabolic Keller-Segel-Stokes system, we show that, by some new energy methods which are different from some known related results, there also exists a particular value. If the initial mass of cells below it, then the corresponding initial-boundary value problem has global-in time and bounded solutions.