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 { n t + u ⋅ ∇ n = Δ n m − ∇ ⋅ ( n ∇ c ) , c t + u ⋅ ∇ c = Δ c − n c , u t + ∇ P = Δ u + 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 > 8 7 and the initial data ( n 0 , c 0 , u 0 ) are sufficiently regular satisfying n 0 > 0 and c 0 > 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 + 217 12 , 2 ] .