Transonic shocks and free boundary problems for the full Euler equations in infinite nozzles

Abstract

We establish the existence, stability, and asymptotic behavior of transonic flows with a transonic shock for the steady, full Euler equations in two-dimensional infinite nozzles of slowly varying cross-sections. Given a smooth incoming flow that is close to a uniform supersonic state at the entrance, we prove that there exists a transonic flow whose infinite downstream smooth subsonic region is separated by a smooth transonic shock from the upstream supersonic flow. The solution is unique within the class of transonic solutions that are close to the background solution. This problem is approached by a free boundary problem in which the transonic shock is formulated as a free boundary. An iteration scheme for the free boundary is developed and its fixed point is shown to exist, which is a solution of the free boundary problem, by combining some delicate estimates for a second-order nonlinear elliptic equation on a Lipschitz domain.