EVENT DETAILS AND ABSTRACT

Title: Stability of Transonic shocks for Euler-Poisson and Euler equations
Speaker: Chunjing Xie
Speaker Info: University of Michigan, Ann Arbor
Brief Description:
Special Note:
Abstract:

In this talk, both structural and dynamical stabilities of steady transonic shock solutions for one-dimensional Euler-Poission system will be discussed. First, a steady transonic shock solution with supersonic backgroumd charge is shown to be structurally stable with respect to small perturbations of the background charge, provided that the electric field is positive at the shock location. Second, any steady transonic shock solution with the supersonic background charge is proved to be dynamically and exponentially stable with respect to small perturbation of the initial data, provided the electric field is not too negative at the shock location. The proof of the first stability result relies on a monotonicity argument for the shock position and the downstream density, and a stability analysis for subsonic and supersonic solutions. The dynamical stability of the steady transonic shock for the Euler-Poisson equations can be transformed to the global well-posedness of a free boundary problem for a quasilinear second order equation with nonlinear boundary conditions. The analysis for the associated linearized problem plays an essential role. This also motivated us to prove the stability of transonic shocks in quasi-one-dimensional nozzles without assumptions on shock strength and the magnitude of geometric source. This is a joint work with Tao Luo, Jeffrey Rauch and Zhouping Xin.