In this talk I will discuss the refraction of shocks on the interface for 2-d steady compressible flow. Particularly, the class of E-H type regular refraction is defined and its global stability of the wave structure is verified. The 2-d steady potential flow equations is employed to describe the motion of the fluid. The stability problem of the E-H type regular refraction can be reduced to a free boundary problem of nonlinear mixed type equations in an unbounded domain. The corresponding linearized problem has similarities to a generalized Tricomi problem of the linear Lavrentiev-Bitsadze mixed type equation, and it can be reduced to a nonlocal boundary value problem of an elliptic system. The later is finally solved by establishing the bijection of the corresponding nonlocal operator in a weighted H\"older space via careful harmonic analysis.
This is a joint work with CHEN Shuxing and HU Dian.