Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Quasi-neutral limit of Nernst-Planck-Navier-Stokes system

Ping Zhang, Yibin Zhang

Abstract

In this paper, we investigate the quasi-neutral limit of Nernst-Planck-Navier-Stokes system in a smooth bounded domain $Ω$ of $\mathbb{R}^d$ for $d=2,3,$ with ``electroneutral boundary conditions" and well-prepared data. We first prove by using modulated energy estimate that the solution sequence converges to the limit system in the norm of $L^\infty((0,T);L^2(Ω))$ for some positive time $T.$ In order to justify the limit in a stronger norm, we need to construct both the initial layers and weak boundary layers in the approximate solutions.

Quasi-neutral limit of Nernst-Planck-Navier-Stokes system

Abstract

In this paper, we investigate the quasi-neutral limit of Nernst-Planck-Navier-Stokes system in a smooth bounded domain of for with ``electroneutral boundary conditions" and well-prepared data. We first prove by using modulated energy estimate that the solution sequence converges to the limit system in the norm of for some positive time In order to justify the limit in a stronger norm, we need to construct both the initial layers and weak boundary layers in the approximate solutions.
Paper Structure (9 sections, 21 theorems, 241 equations)

This paper contains 9 sections, 21 theorems, 241 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and technical lemmas
  3. The Convergence in $L_T^\infty(L^2)$
  4. Analysis of layers
  5. The Convergence in $L^\infty_T(H^2)$
  6. Estimate of $\|(\nabla c_i^S, \varepsilon\Delta \Phi^S,\nabla u^S)\|_{L^\infty_T(L^2)}$.
  7. Estimate of $\|(\partial_tc_i^S,\varepsilon \nabla \partial_t \Phi^S,\partial_t u^S)\|_{L^\infty_T(L^2)}$.
  8. Estimate of $\|\Delta c_i^S\|_{L^\infty_T(L^2)}$.
  9. The proof of Lemma \ref{['estimate of K_i-P_i']}

Key Result

Theorem 1.1

Let $d=2$ or $3$, and the initial data be "well-prepared" i.e. $\rho^\varepsilon(0)=0.$ We assume $(c^\varepsilon_i(0),u^\varepsilon(0), c_i(0),u(0)) \in H^5$. If there exists a positive constant $C>0$ so that then there exist positive constants $M, T>0,$ which depend only on initial data, $\nu,\lambda, \Lambda, W(x)$, $\gamma_i(x), z_i$ and $D_i$ for $i=1,2,$ so that

Theorems & Definitions (46)

  • Remark 1.1
  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.2
  • Remark 1.3
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Remark 2.1
  • ...and 36 more