# Existence of a time periodic solution for the compressible Euler equation with a time periodic outer force

## TL;DR

This paper proves the existence of a time periodic solution for the compressible Euler equations describing supersonic flow under a periodic external force, addressing a fundamental but underexplored problem in conservation laws.

## Contribution

It establishes the existence of time periodic solutions for the compressible Euler equations with periodic outer forces, a novel result in the study of conservation laws.

## Key findings

- Existence of time periodic solutions for the system.
- Application to supersonic flow scenarios.
- Addresses a gap in conservation law research.

## Abstract

We are concerned with a time periodic supersonic flow through a bounded interval. This motion is described by the compressible Euler equation with a time periodic outer force. Our goal in this paper is to prove the existence of a time periodic solution. Although this is a fundamental problem for other equations, it has not been received much attention for the system of conservation laws until now.