Global weak solutions to a 3-dimensional degenerate and singular chemotaxis–Navier–Stokes system with logistic source

Abstract

This paper considers the degenerate and singular chemotaxis–Navier–Stokes system with logistic term nt+u⋅∇n=Δnm−χ∇⋅(n∇c)+κn−μn2,x∈Ω,t>0,ct+u⋅∇c=Δc−nc,x∈Ω,t>0,ut+(u⋅∇)u=Δu+∇P+n∇Φ,∇⋅u=0,x∈Ω,t>0,where Ω⊂R3 is a bounded domain and χ,κ≥0 and m,μ>0. In the above system without fluid environment Jin (2017) showed existence and boundedness of global weak solutions. On the other hand, in the above system with m=1, Lankeit (2016) established global existence of weak solutions. However, the above system with m>0 has not been studied yet. The purpose of this talk is to establish global existence of weak solutions in the chemotaxis–Navier–Stokes system with degenerate diffusion and logistic term.