ARTICLES
Ming-hua YANG, Si-ming HUANG, Jin-yi SUN
In this paper, we study a global zero-relaxation limit problem of the electro-diffusion model arising in electro-hydrodynamics which is the coupled Planck-Nernst-Poisson and Navier-Stokes equations. That is, the paper deals with a singular limit problem of \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}=\Delta \phi^{\epsilon}\nabla\phi^{\epsilon}, \ \ \ &{\rm in}\ \mathbb{R}^{3}\times(0, \infty), \\[5pt] \nabla\cdot u^{\epsilon}=0, \ \ \ &{\rm in}\ \mathbb{R}^{3}\times(0, \infty), \\[5pt] n_t^{\epsilon}+u^{\epsilon}\cdot\nabla n^{\epsilon}-\Delta n^{\epsilon}=-\nabla\cdot(n^{\epsilon}\nabla \phi^{\epsilon}), &{\rm in}\ \mathbb{R}^{3}\times(0, \infty),\\[5pt] c_t^{\epsilon}+u^{\epsilon}\cdot\nabla c^{\epsilon}-\Delta c^{\epsilon}=\nabla\cdot(c^{\epsilon}\nabla\phi^{\epsilon}), &{\rm in}\ \mathbb{R}^{3}\times(0, \infty),\\[5pt] \epsilon^{-1} \phi^{\epsilon}_t= \Delta \phi^{\epsilon}- n^{\epsilon}+ c^{\epsilon}, &{\rm in}\ \mathbb{R}^{3}\times(0, \infty),\\[5pt] (u^{\epsilon}, n^{\epsilon}, c^{\epsilon},\phi^{\epsilon})|_{t=0}= (u_{0}, n_{0}, c_{0},\phi_{0}), &{\rm in}\ \mathbb{R}^{3} \end{array} \right. \end{eqnarray*} involving with a positive, large parameter $\epsilon$. The present work show a case that $(u^{\epsilon}, n^{\epsilon}, c^{\epsilon})$ stabilizes to $(u^{\infty}, n^{\infty}, c^{\infty}):=(u, n, c)$ uniformly with respect to the time variable as $\epsilon\rightarrow+\infty$ with respect to the strong topology in a certain Fourier-Herz space.