Subsonic time-periodic solution to the compressible Euler equations triggered by boundary conditions

Abstract

In this paper, we consider the one-dimensional isentropic compressible Euler equations with source term β(t,x)ρ|u|αu 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. The 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 coefficient β(t,x).