Existence of a time periodic solution for the compressible Euler equation with a time periodic outer force in a bounded interval

TL;DR

This paper proves the existence of time periodic solutions for the compressible Euler equations with an external force in a bounded interval, addressing a gap in understanding of conservation laws without decay estimates.

Contribution

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

Findings

Existence of time periodic solutions proven

Addresses challenges due to lack of decay estimates

Contributes to understanding of fluid dynamics with external forces

Abstract

In the field of differential equations, particularly fluid dynamics, many researchers have shown an interest in the behavior of time periodic solutions. In this paper, we study isentropic gas flow in a bounded interval and apply a time periodic outer force. This motion is described by the compressible Euler equation with the outer force. Our purpose in this paper is to prove the existence of a time periodic solutions. Unfortunately, little is known for the system of conservation laws until now. The problem seems to lie in fact that the equation does not possesses appropriate decay estimates.