A 3D self-consistent chemotaxis-fluid system with nonlinear diffusion

Abstract

In this paper we deal with the initial-boundary value problem for a chemotaxis-fluid model involving a complicated nonlinear coupling term, precisely, the following self-consistent system with porous medium diffusion{nt+u⋅∇n=Δnm−∇⋅(n∇c)+∇⋅(n∇ϕ),x∈Ω,t>0,ct+u⋅∇c=Δc−nc,x∈Ω,t>0,ut+∇P=Δu−n∇ϕ+n∇c,x∈Ω,t>0,∇⋅u=0,x∈Ω,t>0, under no-flux boundary conditions in a bounded domain Ω⊂R3 with smooth boundary.One of the novelties here is that both the effect of gravity (potential force) on cells and the effect of the chemotactic force on fluid is considered, which leads to the stronger coupling than the most-studied chemotaxis-fluid model for swimming bacteria proposed in [28]. We shall establish the existence of global weak solutions whenever m>43, thereby supplementing previously gained knowledge on its two-dimensional version.