Critical mass phenomenon in a chemotaxis fluid system

Abstract

In this paper, we study the following parabolic–parabolic–elliptic attraction–repulsion chemotaxis fluid system driven by a friction force n∇(χ1c−χ2v): (0.1)nt+u⋅∇n=Δn−∇⋅(χ1n∇c)+∇⋅(χ2n∇v),x∈Ω,t>0,ct=Δc−λ1c+n,x∈Ω,t>0,0=Δv−λ2v+n,x∈Ω,t>0,ut+(u⋅∇)u+∇P=Δu+n∇(χ1c−χ2v),∇⋅u=0,x∈Ω,t>0,∂n∂ν=∂c∂ν=∂v∂ν=0,u=0,x∈∂Ω,t>0,n(x,0)=n0(x),c(x,0)=c0(x),u(x,0)=u0(x),x∈Ω,in a bounded domain Ω⊂R2 with smooth boundary. With the help of a free energy functional, it is proved that (0.1) admits a global classical solution with arbitrary large initial data in the repulsive dominant or balance case χ1≤χ2. Moreover, there exists a critical mass phenomenon in the attractive dominant case χ1>χ2. Specifically, when the initial mass (χ1−χ2)∫Ωn0(x)dx≤4π, the solution exists globally; while if the initial mass (χ1−χ2)∫Ωn0(x)dx>4π and (χ1−χ2)∫Ωn0(x)dx∉{4πm:m∈N+}, the solution may blow up. This work extends the conclusions of the critical mass phenomenon in the Patlak–Keller–Segel-Navier–Stokes system (Gong and He, 2021) to the parabolic–parabolic–elliptic attraction–repulsion chemotaxis fluid system.