Global existence and stability of time-periodic solution to isentropic compressible Euler equations with source term

TL;DR

This paper proves the global existence and time-periodic stability of smooth solutions to one-dimensional isentropic compressible Euler equations with a source term, under small perturbations around a specific flow.

Contribution

It establishes the existence of global smooth solutions and their time-periodic nature for the Euler equations with a source term, using wave decomposition and a-priori estimates.

Findings

Global smooth solutions exist under small perturbations.

Solutions are proven to be time-periodic.

The analysis applies to flows near the supersonic Fanno flow.

Abstract

In this paper, we study the initial-boundary value problem of one-dimensional isentropic compressible Euler equations with the source term $\beta\rho|u|^{\alpha}u$. By means of wave decomposition and the uniform a-priori estimates, we prove the global existence of smooth solutions under small perturbations around the supersonic Fanno flow. Then, by Gronwall's inequality, we get the smooth solution is time-periodic.