Jian-xiang WAN, Hai-ping ZHONG
The paper deals with a Cauchy problem for the chemotaxis system with the effect of fluid \begin{eqnarray*}\label{1.2} \left\{ \begin{array}{ll} u_t^{\epsilon}+u^{\epsilon}\cdot\nabla u^{\epsilon}-\Delta u^{\epsilon}+\nabla\mathbf{P}^{\epsilon}= n^{\epsilon}\nabla c^{\epsilon}, \ \ \ &{\rm in}\ \mathbb{R}^{d}\times (0, \infty), \\[6pt] \nabla\cdot u^{\epsilon}=0, \ \ \ &{\rm in}\ \mathbb{R}^{d}\times (0, \infty), \\[6pt] n_t^{\epsilon}+u^{\epsilon}\cdot\nabla n^{\epsilon}-\Delta n^{\epsilon}=-\nabla\cdot(n^{\epsilon}\nabla c^{\epsilon}), &{\rm in}\ \mathbb{R}^{d}\times (0, \infty),\\[6pt] \frac{1}{\epsilon}c_t^{\epsilon}-\Delta c^{\epsilon}= n^{\epsilon}, &{\rm in}\ \mathbb{R}^{d}\times (0, \infty),\\[6pt] (u^{\epsilon}, n^{\epsilon}, c^{\epsilon})|_{t=0}= (u_{0}, n_{0}, c_{0}), &{\rm in}\ \mathbb{R}^{d},\\[6pt] \end{array} \right. \end{eqnarray*} where $d\geq2$. It is known that for each $\epsilon>0$ and all sufficiently small initial data $(u_{0},n_{0},c_{0})$ belongs to certain Fourier space, the problem possesses a unique global solution $(u^{\epsilon},n^{\epsilon},c^{\epsilon})$ in Fourier space. The present work asserts that these solutions stabilize to $(u^{\infty},n^{\infty},c^{\infty})$ as $\epsilon^{-1}\rightarrow 0$. Moreover, we show that $c^{\epsilon}(t)$ has the initial layer as $\epsilon^{-1}\rightarrow 0$. As one expects its limit behavior maybe give a new viewlook to understand the system.