Locally bounded global solutions in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion

Abstract

This paper deals with a boundary-value problem in three-dimensional smoothly bounded domains for a coupled chemotaxis-Stokes system generalizing the prototype{nt+u⋅∇n=Δnm−∇⋅(n∇c),ct+u⋅∇c=Δc−nc,ut+∇P=Δu+n∇ϕ,∇⋅u=0, which describes the motion of oxygen-driven swimming bacteria in an incompressible fluid.It is proved that global weak solutions exist whenever m>87 and the initial data (n0,c0,u0) are sufficiently regular satisfying n0>0 and c0>0. This extends a recent result by Di Francesco, Lorz and Markowich [M. Di Francesco, A. Lorz, P.A. Markowich, Chemotaxis–fluid coupled model for swimming bacteria with nonlinear diffusion: Global existence and asymptotic behavior, Discrete Contin. Dyn. Syst. Ser. A 28 (2010) 1437–1453] which asserts global existence of weak solutions under the constraint m∈[7+21712,2].