2018/02/24 by Kurima, Shunsuke, Mizukami, Masaaki
#35Q35 #Analysis of PDEs (math.AP) #FOS: Mathematics #Primary: 35K55 #Secondary: 92C17
paper · doi:10.48550/arxiv.1802.08807
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 (J. Differential Equations, 2017) showed existence and boundedness of global weak solutions. On the other hand, in the above system with m=1, Lankeit (Math. Models Methods Appl. Sci., 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.