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

TL;DR

This paper proves that small, non-trivial, compact solutions do not exist for the 1D Euler-Poisson system, using virial estimates to show decay properties of solutions.

Contribution

It establishes the non-existence of small compact solutions for the 1D Euler-Poisson equations, a result not previously known.

Findings

Small non-trivial compact solutions do not exist in 1D Euler-Poisson.

Virial estimates imply decay of bounded small solutions.

The proof relies on local in space average decay analysis.

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.