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A note on the non-existence of small non-trivial compact solutions for Euler-Poisson equation in 1D

Masaya Maeda, Tetsu Mizumachi

Abstract

In this short note, we prove the non-existence of slow and fast small nontrivial compact solutions for the Euler-Poisson system in $1$D. The proof is based on the virial estimate which provides local in space average decay of bounded small solutions.

A note on the non-existence of small non-trivial compact solutions for Euler-Poisson equation in 1D

Abstract

In this short note, we prove the non-existence of slow and fast small nontrivial compact solutions for the Euler-Poisson system in D. The proof is based on the virial estimate which provides local in space average decay of bounded small solutions.
Paper Structure (4 sections, 15 theorems, 89 equations)

This paper contains 4 sections, 15 theorems, 89 equations.

Table of Contents

  1. Euler-Poisson equation
  2. Preliminary
  3. Proof of Theorem \ref{['thm:smallslow']}
  4. Proof of Theorem \ref{['thm:smallfast']}

Key Result

Theorem 1.3

For any $\varepsilon>0$, there exists $\delta>0$ such that if $(n,u,\phi)$ is an $L^2$-compact solution satisfying and conserves the energy $E$, then $(n,u,\phi)=0$.

Theorems & Definitions (33)

  • Definition 1.1
  • Remark 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • ...and 23 more