Global existence and stability of subsonic time-periodic solution to the damped compressible Euler equations in a bounded domain

TL;DR

This paper proves the existence and stability of a unique subsonic time-periodic smooth solution to the damped compressible Euler equations in a bounded domain with periodic boundary conditions, using linear iterative methods.

Contribution

It introduces a novel approach to establish the existence and stability of subsonic periodic solutions without small damping assumptions.

Findings

Existence of a unique subsonic time-periodic smooth solution.

Stability of the periodic solution under small perturbations.

No small assumptions on the damping coefficient.

Abstract

In this paper, we consider the one-dimensional isentropic compressible Euler equations with source term $\beta(t,x)\rho|u|^{\alpha}u$ in a bounded domain, which can be used to describe gas transmission in a nozzle.~The model is imposed a subsonic time-periodic boundary condition.~Our main results reveal that the time-periodic boundary can trigger an unique subsonic time-periodic smooth solution and this unique periodic solution is stable under small perturbations on initial and boundary data.~To get the existence of subsonic time-periodic solution, we use the linear iterative skill and transfer the boundary value problem into two initial value ones by using the hyperbolic property of the system. Then the corresponding linearized system can be decoupled.~The uniqueness is a direct by-product of the stability. There is no small assumptions on the damping coefficient.