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Stability of Stationary Solutions to the Nonisentropic Euler-Poisson System in a Perturbed Half Space

Mingjie Li, Masahiro Suzuki

Abstract

The main concern of this paper is to mathematically investigate the formation of a plasma sheath near the surface of nonplanar walls. We study the existence and asymptotic stability of stationary solutions for the nonisentropic Euler-Poisson equations in a domain of which boundary is drawn by a graph, by employing a space weighted energy method. Moreover, the convergence rate of the solution toward the stationary solution is obtained, provided that the initial perturbation belongs to the weighted Sobolev space. Because the domain is the perturbed half space, we first show the time-global solvability of the nonisentropic Euler-Poisson equations, then construct stationary solutions by using the time-global solutions.

Stability of Stationary Solutions to the Nonisentropic Euler-Poisson System in a Perturbed Half Space

Abstract

The main concern of this paper is to mathematically investigate the formation of a plasma sheath near the surface of nonplanar walls. We study the existence and asymptotic stability of stationary solutions for the nonisentropic Euler-Poisson equations in a domain of which boundary is drawn by a graph, by employing a space weighted energy method. Moreover, the convergence rate of the solution toward the stationary solution is obtained, provided that the initial perturbation belongs to the weighted Sobolev space. Because the domain is the perturbed half space, we first show the time-global solvability of the nonisentropic Euler-Poisson equations, then construct stationary solutions by using the time-global solutions.
Paper Structure (16 sections, 17 theorems, 102 equations)

This paper contains 16 sections, 17 theorems, 102 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Reformulation and Preliminary
  4. Reformulation
  5. Preliminary
  6. Time-global solvability
  7. Elliptic estimates
  8. Basic estimate
  9. Higher order estimate
  10. Completion of a priori estimate
  11. Construction of stationary solutions
  12. Time-periodic solutions
  13. Uniqueness
  14. Existence
  15. Stationary solutions
  16. ...and 1 more sections

Key Result

Lemma 2.1

Let $u_+$ and $\theta_{+}$ satisfy Bohm1. There exist a constant $\delta>0$ such that if $|\phi_b| < \delta$, then the problem sp0 has a unique monotone solution $(\tilde{\rho} ,\tilde{u},\tilde{\theta}, \tilde{\phi}) \in {\cal B}^{\infty}(\overline{\mathbb{R}_{+}})$. Moreover, it satisfies where $\alpha<1$ and $C$ are positive constants independent of $\phi_b$.

Theorems & Definitions (28)

  • Lemma 2.1: DYZ1
  • Theorem 2.2
  • Theorem 2.3
  • Remark 2.4
  • Theorem 3.1
  • Theorem 3.2
  • Lemma 3.3: MM1
  • Theorem 4.1
  • Lemma 4.2
  • Proposition 4.3
  • ...and 18 more