Global bounded weak solutions in a two-dimensional Keller-Segel-Navier-Stokes system with indirect signal production and nonlinear diffusion

Abstract

This paper considers the coupled quasilinear Keller–Segel–Navier–Stokes system with indirect signal production and nonlinear diffusion(⁎){nt+u⋅∇n=Δnm−∇⋅(n∇v),x∈Ω,t>0,vt+u⋅∇v=Δv−v+w,x∈Ω,t>0,wt+u⋅∇w=Δw−w+n,x∈Ω,t>0,ut+(u⋅∇)u+∇P=Δu+n∇ϕ,x∈Ω,t>0,∇⋅u=0,x∈Ω,t>0 under no-flux boundary conditions for n, v and w and no-slip boundary condition for u in a bounded domain Ω⊂R2 with smooth boundary, where ϕ∈W2,∞(Ω) and m>0. If m>1, then for any sufficiently smooth initial data, there exists at least one globally defined weak solution which is bounded for the corresponding initial-boundary value problem of system (⁎).