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 $$ \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}, \boldsymbol{u}^{\kappa}_{t}+\kappa (\boldsymbol{u}^{\kappa}\cdot )\boldsymbol{u}^{\kappa}=\Delta \boldsymbol{u}^{\kappa}+\nabla{P}^{\kappa}+n^{\kappa}\nabla \phi \boldsymbol{u}^{\kappa}=0, & x\in \end{cases} { n t κ + u ⋅ ∇ = Δ − χ ( log c ) f , x ∈ Ω > 0 P ϕ in a bounded convex domain $\Omega \subseteq \mathbb{R}^{2}$ ⊆ R 2 smooth boundary, where $\kappa \in \mathbb{R}$ $f(s)=\mu _{1} s-\mu _{2} s^{\lambda}$ s μ 1 λ $\lambda >1$ $\phi :\Omega \rightarrow : → is given potential second-order partial derivatives. When chemotaxis χ satisfies appropriate conditions, it proved that unique global classical solutions $(n^{\kappa},c^{\kappa},\boldsymbol{u}^{\kappa})$ will stabilize to $(n^{0},c^{0},\boldsymbol{u}^{0})$ as 0$ .