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The existence and asymptotic stability of Plasma-Sheaths to the full Euler-Poisson system

Yao Lei, Yin Haiyan, Zhu Mengmeng

Abstract

The main concern of this paper is to study large-time behavior of the sheath to the full Euler-Poisson system. As is well known, the monotone stationary solution under the Bohm criterion can be referred to as the sheath which is formed by interactions of plasma with wall. So far, the existence and asymptotic stability of stationary solutions in one-dimensional half space to the full Euler-Poisson system have been proved in \cite{DYZ2021}. In the present paper, we extend the results in \cite{DYZ2021} to $N$-dimensional ($N$=1,2,3) half space. By assuming that the velocity of the positive ion satisfies the Bohm criterion at the far field, we establish the global unique existence and the large time asymptotic stability of the sheath in some weighted Sobolev spaces by weighted energy method. Moreover, the time-decay rates are also obtained. A key different point from \cite{DYZ2021} is to derive some boundary estimates on the derivative of the potential in the $x_1$-direction.

The existence and asymptotic stability of Plasma-Sheaths to the full Euler-Poisson system

Abstract

The main concern of this paper is to study large-time behavior of the sheath to the full Euler-Poisson system. As is well known, the monotone stationary solution under the Bohm criterion can be referred to as the sheath which is formed by interactions of plasma with wall. So far, the existence and asymptotic stability of stationary solutions in one-dimensional half space to the full Euler-Poisson system have been proved in \cite{DYZ2021}. In the present paper, we extend the results in \cite{DYZ2021} to -dimensional (=1,2,3) half space. By assuming that the velocity of the positive ion satisfies the Bohm criterion at the far field, we establish the global unique existence and the large time asymptotic stability of the sheath in some weighted Sobolev spaces by weighted energy method. Moreover, the time-decay rates are also obtained. A key different point from \cite{DYZ2021} is to derive some boundary estimates on the derivative of the potential in the -direction.
Paper Structure (6 sections, 19 theorems, 175 equations)

This paper contains 6 sections, 19 theorems, 175 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Energy estimates for the degenerate case
  4. A priori estimates for the degenerate case
  5. Energy estimates for the nondegenerate case
  6. A priori estimates for the nondegenerate case

Key Result

Lemma 1.1

(see DYZ2021). Consider the boundary-value problem C8$-$C9. (i) Let $u_\infty$ be a constant satisfying Then the stationary problem C8$-$C9 has a unique monotone solution $(\tilde{n}, \tilde{u}, \tilde{T},\tilde{\phi})(x_1)$ verifying if and only if the boundary data $\phi_b$ satisfies conditions where $c_\infty=\left(\frac{mu_\infty^2}{\gamma RT_\infty}\right)^{\frac{1}{\gamma+1}}$ is the onl

Theorems & Definitions (32)

  • Lemma 1.1
  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • Proposition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • ...and 22 more