Mathematical Research Letters

Volume 16 (2009)

Number 5

A free boundary value problem for the full Euler system and 2-d transonic shock in a large variable nozzle

Pages: 777 – 796



Jun Li (The Institute of Mathematical Sciences, CUHK, Hong Kong)

Zhouping Xin (The Institute of Mathematical Sciences, CUHK, Hong Kong)

Huicheng Yin (Department of Mathematics & IMS, Nanjing University, China)


In this paper, we establish the existence and uniqueness of a transonic shock solution to the full steady compressible Euler system in a class of de Laval nozzles with a large straight divergent part when a given variable exit pressure lies in a suitable range. Thus, for this class of nozzles, we have solved the transonic shock problem posed by Courant-Friedrichs in Section 147 of [5]. By introducing a new elaborate iteration scheme, we are able to solve this boundary value problem for a coupled elliptic-hyperbolic system with a free boundary without some stringent requirements in the previous studies. One of the key ingredients in this approach is to solve a boundary value problem for a first order linear system with nonlocal terms and a free parameter.

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