Steady transonic shocks and free boundary problems for the Euler equations in infinite cylinders

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 infi-nite 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 constant states. 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. Résumé: Nous établissons l’existence, la stabilit e ́ et le comportement asymptotique des écoulements transoniques avec un choc transonique pour les équations d’Euler complètes et indépendantes du temps dans des tuyères bidimensionelles, infinies et ayant des coupes transversales qui varient lentement. Etant donne ́ un écoulement entrant régulier qui es