Dynamical Stability of Transonic Shock Solutions to Non-Isentropic Euler Equations

Abstract

In this paper, we investigate the dynamical stability of transonic shock solutions for the full compressible Euler system in a two dimensional nozzle with a symmetric divergent part. Building upon the existence and uniqueness results for steady symmetric transonic shock solutions to the non-isentropic Euler system established in [Z.P. Xin and H.C. Yin, The transonic shock in a nozzle, 2-D and 3-D complete Euler systems, J. Differential Equations 245 (2008)], we prove the dynamical stability of the transonic shock solutions under small perturbations. More precisely, if the initial unsteady transonic flow is located in the symmetric divergent part of the nozzle and the flow is a symmetric small perturbation of the steady transonic flow, we use the characteristic method to establish the dynamical stability.