Asymptotic profile of a two-dimensional chemotaxis--Navier--Stokes system with singular sensitivity and logistic source

2020/12/24 by Pang, Peter Y. H., Wang, Yifu, Yin, Jingxue
#35B40 #35K55 #35Q30 #35Q92 #76D05 #92C17 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2012.13116

Abstract

The chemotaxis--Navier--Stokes system \n_t+u⋅ ∇ n=\triangle n-χ∇\cdotp (\frac n c∇ c)+n(r-μn), c_t+u⋅ ∇ c=\triangle c-nc, u_t+ (u⋅ ∇) u=Δu+∇ P+n∇ϕ, ∇⋅ u=0,. is considered in a bounded smooth domain Ω⊂ ℝ², where ϕ∈ W^1,∞(Ω), χ>0, r∈ ℝ and μ> 0 are given parameters. It is shown that there exists a value μ_*(Ω,χ, r)≥ 0 such that whenever μ>μ_*(Ω,χ, r), the global-in-time classical solution to the system is uniformly bounded with respect to x∈ Ω. Moreover, for the case r>0, (n,c,\frac |∇ c|c,u) converges to (\frac r μ,0,0,0) in L^∞(Ω)× L^∞(Ω)× L^p(Ω)× L^∞(Ω) for any p>1 exponentially as t→ ∞, while in the case r=0, (n,c,\frac |∇ c|c,u) converges to (0,0,0,0) in (L^∞(Ω))⁴ algebraically. To the best of our knowledge, these results provide the first precise information on the asymptotic profile of solutions in two dimensions.

Related

• Very mild diffusion enhancement and singular sensitivity: Existence of bounded weak solutions in a two-dimensional chemotaxis-Navier--Stokes system
• Small-mass solutions in a two-dimensional logarithmic chemotaxis-Navier-Stokes system with indirect nutrient consumption
• Long-term behaviour in a chemotaxis-fluid system with logistic source
• Global classical small-data solutions for a three-dimensional chemotaxis Navier-Stokes system involving matrix-valued sensitivities
• Generalized solutions to a chemotaxis-Navier-Stokes system with arbitrary superlinear degradation