Small-convection limit for two-dimensional chemotaxis-Navier–Stokes system with logarithmic sensitivity and logistic-type source

Abstract

In this paper, we consider the small-convection limit of chemotaxis-Navier–Stokes system with logarithmic sensitivity and logistic-type source {ntκ+uκ⋅∇nκ=Δnκ−χ∇⋅(nκ∇logcκ)+f(nκ),x∈Ω,t>0,ctκ+uκ⋅∇cκ=Δcκ−cκ+nκ,x∈Ω,t>0,utκ+κ(uκ⋅∇)uκ=Δuκ+∇Pκ+nκ∇ϕ,x∈Ω,t>0,∇⋅uκ=0,x∈Ω,t>0,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \ extstyle\\begin{cases} n^{\\kappa}_{t}+\\boldsymbol{u}^{\\kappa}\\cdot \ abla{n}^{\\kappa}= \\Delta{n}^{\\kappa}-\\chi \ abla \\cdot ({n^{\\kappa}}\ abla \\log{c^{\\kappa}})+f(n^{\\kappa}), &x\\in \\Omega , t>0, \\\\ c^{\\kappa}_{t}+\\boldsymbol{u}^{\\kappa}\\cdot \ abla{c}^{\\kappa}=\\Delta{c}^{\\kappa}-c^{\\kappa}+n^{\\kappa}, &x\\in \\Omega , t>0, \\\\ \\boldsymbol{u}^{\\kappa}_{t}+\\kappa (\\boldsymbol{u}^{\\kappa}\\cdot \ abla )\\boldsymbol{u}^{\\kappa}=\\Delta \\boldsymbol{u}^{\\kappa}+\ abla{P}^{\\kappa}+n^{\\kappa}\ abla \\phi , &x\\in \\Omega , t>0, \\\\ \ abla \\cdot \\boldsymbol{u}^{\\kappa}=0, & x\\in \\Omega , t>0, \\end{cases} $$\\end{document} in a bounded convex domain Omega subseteq mathbb{R}^{2} with smooth boundary, where kappa in mathbb{R}, f(s)=mu _{1} s-mu _{2} s^{lambda}, lambda >1, and phi :Omega rightarrow mathbb{R} is a given smooth potential with second-order partial derivatives. When the chemotaxis sensitivity χ satisfies the appropriate conditions, it is proved that the unique global classical solutions (n^{kappa},c^{kappa},boldsymbol{u}^{kappa}) will stabilize to (n^{0},c^{0},boldsymbol{u}^{0}) as kappa rightarrow 0.