Free Boundary Problem related to Euler-Poisson system

One dimensional analysis of Euler-Poisson system shows that when incoming supersonic flow is fixed,

transonic shock can be represented as a monotone function of exit pressure.

From this observation, we expect well-posedness of transonic shock problem for Euler-Poisson system

when exit pressure is prescribed in a proper range.

In this talk, I will present recent progress on transonic shock problem for Euler-Poisson system,

which is formulated as a free boundary problem with mixed type PDE system.

This talk is based on collaboration with Ben Duan, Chujing Xie and Jingjing Xiao