Boundedness of weak solutions in a 3D chemotaxis-Stokes system with nonlinear doubly degenerate diffusion

Abstract

In this paper, the following incompressible chemotaxis-Stokes system with nonlinear doubly degenerate diffusion is considered in a smooth bounded domain Ω⊂R3:{nt+u⋅∇n=∇⋅(|∇nm|p−2∇nm)−∇⋅(n∇c),x∈Ω,t>0,ct+u⋅∇c=Δc−nc,x∈Ω,t>0,ut=Δu+∇P+n∇ϕ,x∈Ω,t>0,∇⋅u=0,x∈Ω,t>0, where the potential function ϕ∈W2,∞(Ω) is given. For homogeneous boundary conditions of Neumann type for n and c, and of Dirichlet type for u, it is proved that global bounded weak solutions exist for suitable regular initial data whenever 8mp−8m+3p>15, p≥2 and m≥1.