Global Zero-relaxation Limit Problem of the Electro-diffusion Model Arising in Electro-Hydrodynamics

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 \(\left\{ \begin{gathered}\begin{array}{*{20}{c}}{u_t^\varepsilon+ {u^\varepsilon } \cdot \nabla {u^\varepsilon } - \Delta {u^\varepsilon } + \nabla {P^\varepsilon } = \Delta {\phi ^\varepsilon }\nabla {\phi ^\varepsilon },}&{in{\text{ }}{\mathbb{R}^3} \times (0,\infty )} \\ {\nabla\cdot {u^\varepsilon } = 0,}&{in{\text{ }}{\mathbb{R}^3} \times (0,\infty )} \end{array} \hfill \\\begin{array}{*{20}{c}}{n_t^\varepsilon+ {u^\varepsilon } \cdot \nabla {n^\varepsilon } - \Delta {n^\varepsilon } =- \nabla\cdot ({n^\varepsilon }\nabla {\phi ^\varepsilon }),}&{in{\text{ }}{\mathbb{R}^3} \times (0,\infty )} \\ {c_t^\varepsilon+ {u^\varepsilon } \cdot \nabla {c^\varepsilon } - \Delta {c^\varepsilon } = \nabla\cdot ({c^\varepsilon }\nabla {\phi ^\varepsilon }),}&{in{\text{ }}{\mathbb{R}^3} \times (0,\infty )} \end{array} \hfill \\\begin{array}{*{20}{c}}{{\varepsilon ^{ - 1}}\phi _t^\varepsilon= \Delta {\phi ^\varepsilon } - {n^\varepsilon } + {c^\varepsilon },}&{in{\text{ }}{\mathbb{R}^3} \times (0,\infty )} \\ {({u^\varepsilon },{n^\varepsilon },{c^\varepsilon },{\phi ^\varepsilon })\left| {_{t = 0 = ({u_0},{n_0},{c_0},{\phi _0})},} \right.}&{in{\text{ }}{\mathbb{R}^3}} \end{array} \hfill \\ \end{gathered}\right.\) $\{\begin{array}{c}\begin{array}{cc}{u}_t^{\epsilon }+u^{\epsilon }\cdot \mathrm{\nabla }{u}^{\epsilon }-\mathrm{\Delta }{u}^{\epsilon }+\mathrm{\nabla }{P}^{\epsilon }=\mathrm{\Delta }{\varphi }^{\epsilon }\mathrm{\nabla }{\varphi }^{\epsilon },& in{\mathbb{R}}^3\times (0,\mathrm{\infty })\\ \mathrm{\nabla }\cdot u^{\epsilon }=0,& in{\mathbb{R}}^3\times (0,\mathrm{\infty })\end{array}\\ \begin{array}{cc}{n}_t^{\epsilon }+u^{\epsilon }\cdot \mathrm{\nabla }{n}^{\epsilon }-\mathrm{\Delta }{n}^{\epsilon }=-\mathrm{\nabla }\cdot (n^{\epsilon }\mathrm{\nabla }{\varphi }^{\epsilon }),& in{\mathbb{R}}^3\times (0,\mathrm{\infty })\\ c_t^{\epsilon }+u^{\epsilon }\cdot \mathrm{\nabla }{c}^{\epsilon }-\mathrm{\Delta }{c}^{\epsilon }=\mathrm{\nabla }\cdot (c^{\epsilon }\mathrm{\nabla }{\varphi }^{\epsilon }),& in{\mathbb{R}}^3\times (0,\mathrm{\infty })\end{array}\\ \begin{array}{cc}{\epsilon }^{-1}{\varphi }_t^{\epsilon }=\mathrm{\Delta }{\varphi }^{\epsilon }-n^{\epsilon }+c^{\epsilon },& in{\mathbb{R}}^3\times (0,\mathrm{\infty })\\ (u^{\epsilon },n^{\epsilon },c^{\epsilon },{\varphi }^{\epsilon })|{}_{t=0=(u_0,n_0,c_0,{\varphi }_0)},& in{\mathbb{R}}^3\end{array}\end{array}$ involving with a positive, large parameter ϵ. The present work show a case that (u^ϵ, n^ϵ, c^ϵ) stabilizes to (u^∞, n^∞, c^∞):= (u, n, c) uniformly with respect to the time variable as ϵ → + ∞ with respect to the strong topology in a certain Fourier-Herz space.