In this paper, we consider the two‐dimensional chemotaxis‐Navier‐Stokes system with singular sensitivity

$$\begin{array}{c}\hfill \left\{\begin{array}{ll}{n}_t+u·\nabla n=\mathrm{\Delta }n-\chi \nabla ·(\frac{n}{c}\nabla c),& x\in \mathrm{\Omega },t>0,\\ c_t+u·\nabla c=\mathrm{\Delta }c-c+n,& x\in \mathrm{\Omega },t>0,\\ u_t+\kappa (u·\nabla )u=\mathrm{\Delta }u+\nabla P+n\nabla \varphi ,& x\in \mathrm{\Omega },t>0,\\ \nabla ·u=0,& x\in \mathrm{\Omega },t>0\end{array}\right.\end{array}$$

in a bounded convex domain Ω ⊂ R² with smooth boundary, with κ ∈ R and a given smooth potential ϕ : Ω → R. It is known that for each κ ∈ [0, 1) and 0 < χ < 1 this problem possesses a unique global classical solution (n^κ, c^κ, u^κ). Our main result asserts that under the assumption of 0 < χ < 1, (n^κ, c^κ, u^κ) stabilizes to (n⁰, c⁰, u⁰) with an explicit rate and a time dependent coefficient as κ → 0₊.

中文翻译：

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具有奇异灵敏度的二维趋化Navier-Stokes系统中的小对流极限

在本文中，我们考虑具有奇异灵敏度的二维化学趋化-Navier-Stokes系统

$$\begin{array}{c}\hfill \left\{\begin{array}{ll}{ñ}_{Ť}+ü·\nabla ñ=\mathrm{\Delta }ñ-\chi \nabla ·（\frac{ñ}{C}\nabla C），& X\in \mathrm{\Omega }，Ť>0，\\ C_{Ť}+ü·\nabla C=\mathrm{\Delta }C-C+ñ，& X\in \mathrm{\Omega }，Ť>0，\\ {ü}_{Ť}+\kappa （ü·\nabla ）ü=\mathrm{\Delta }ü+\nabla P+ñ\nabla \varphi ，& X\in \mathrm{\Omega }，Ť>0，\\ \nabla ·ü=0，& X\in \mathrm{\Omega }，Ť>0\end{array}\right.\end{array}$$

在一个有界域凸Ω⊂  - [R ²与平滑边界，与κ  ＆Element;  [R和给定的光滑电位φ  ：Ω→  [R 。已知的是，对于每个κ  ＆Element; [0，1）和0 <  χ  <1该问题具有独特的全球经典溶液（Ñ ^κ，  Ç ^κ，  Ú ^κ）。我们的主要结果断言的假设下0 <  χ  <1 ，（ Ñ ^κ，  Ç ^κ， Ù ^κ）稳定到（Ñ ⁰，  Ç ⁰，  Ü ⁰）具有显式速率和随时间变化的系数作为κ  →交通0 ₊。