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

In this paper, we consider the small-convection limit of chemotaxis-Navier–Stokes system with logarithmic sensitivity and logistic-type source $$ \textstyle\begin{cases} n^{\kappa}_{t}+\boldsymbol{u}^{\kappa}\cdot \nabla{n}^{\kappa}= \Delta{n}^{\kappa}-\chi \nabla \cdot ({n^{\kappa}}\nabla \log{c^{\kappa}})+f(n^{\kappa}), &x\in \Omega , t>0, \\ c^{\kappa}_{t}+\boldsymbol{u}^{\kappa}\cdot \nabla{c}^{\kappa}=\Delta{c}^{\kappa}-c^{\kappa}+n^{\kappa}, &x\in \Omega , t>0, \\ \boldsymbol{u}^{\kappa}_{t}+\kappa (\boldsymbol{u}^{\kappa}\cdot \nabla )\boldsymbol{u}^{\kappa}=\Delta \boldsymbol{u}^{\kappa}+\nabla{P}^{\kappa}+n^{\kappa}\nabla \phi , &x\in \Omega , t>0, \\ \nabla \cdot \boldsymbol{u}^{\kappa}=0, & x\in \Omega , t>0, \end{cases} $$ { n t κ + u κ ⋅ ∇ n κ = Δ n κ − χ ∇ ⋅ ( n κ ∇ log c κ ) + f ( n κ ) , x ∈ Ω , t > 0 , c t κ + u κ ⋅ ∇ c κ = Δ c κ − c κ + n κ , x ∈ Ω , t > 0 , u t κ + κ ( u κ ⋅ ∇ ) u κ = Δ u κ + ∇ P κ + n κ ∇ ϕ , x ∈ Ω , t > 0 , ∇ ⋅ u κ = 0 , x ∈ Ω , t > 0 , in a bounded convex domain $\Omega \subseteq \mathbb{R}^{2}$ Ω ⊆ R 2 with smooth boundary, where $\kappa \in \mathbb{R}$ κ ∈ R , $f(s)=\mu _{1} s-\mu _{2} s^{\lambda}$ f ( s ) = μ 1 s − μ 2 s λ , $\lambda >1$ λ > 1 , and $\phi :\Omega \rightarrow \mathbb{R}$ ϕ : Ω → 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 $(n^{\kappa},c^{\kappa},\boldsymbol{u}^{\kappa})$ ( n κ , c κ , u κ ) will stabilize to $(n^{0},c^{0},\boldsymbol{u}^{0})$ ( n 0 , c 0 , u 0 ) as $\kappa \rightarrow 0$ κ → 0 .

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