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The relaxation limit of bipolar fluid models

Nuno J. Alves, Athanasios E. Tzavaras

Abstract

This work establishes the relaxation limit from the bipolar Euler-Poisson system to the bipolar drift-diffusion system, for data so that the latter has a smooth solution. A relative energy identity is developed for the bipolar fluid system and is used to show that a dissipative weak solution of the bipolar Euler-Poisson system converges in the high-friction regime to a strong and bounded away from vacuum solution of the bipolar drift-diffusion system.

The relaxation limit of bipolar fluid models

Abstract

This work establishes the relaxation limit from the bipolar Euler-Poisson system to the bipolar drift-diffusion system, for data so that the latter has a smooth solution. A relative energy identity is developed for the bipolar fluid system and is used to show that a dissipative weak solution of the bipolar Euler-Poisson system converges in the high-friction regime to a strong and bounded away from vacuum solution of the bipolar drift-diffusion system.

Paper Structure

This paper contains 10 sections, 9 theorems, 144 equations.

Table of Contents

  1. Introduction
  2. Bipolar fluid models
  3. Bipolar drift-diffusion
  4. Bipolar Euler-Poisson
  5. Relative energy identity for the bipolar fluid system
  6. Statement of the main result
  7. Convergence in the relaxation limit
  8. Auxiliary results
  9. Derivation of the relative energy inequality
  10. Bounds in terms of the relative energy

Key Result

Theorem \oldthetheorem

Let $(\rho,\rho u,n,n v)$, with $\phi = N*(\rho - n)$, be a dissipative weak solution of (BEP) with $\gamma_1,\gamma_2 \geq 2 - \frac{1}{d},$ and let $(\bar{\rho}, \bar{\rho} \bar{u}, \bar{n}, \bar{n} \bar{v})$, with $\bar{\phi}= N*(\bar{\rho}-\bar{n})$, be a strong and bounded away from vacuum solu Therefore if $\Psi(0) \to 0$ as $\varepsilon \to 0$, then $\Psi(t) \to 0$ as $\varepsilon \to 0$ fo

Theorems & Definitions (17)

  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • Theorem \oldthetheorem
  • Proposition \oldthetheorem
  • Proposition \oldthetheorem
  • proof
  • Lemma \oldthetheorem
  • Proposition \oldthetheorem
  • proof
  • Lemma \oldthetheorem
  • ...and 7 more