We establish the existence and uniqueness of transonic flows with a transonic shock through a twodimensional nozzle of slowly varying cross-sections. The transonic flow is governed by the steady, full Euler equations. Given an incoming smooth flow that is close to a constant supersonic state (i.e., smooth Cauchy data) at the entrance and the subsonic condition with nearly horizontal velocity at the exit of the nozzle, we prove that there exists a transonic flow whose downstream smooth subsonic region is separated by a smooth transonic shock from the upstream supersonic flow. This problem is approached by a onephase free boundary problem in which the transonic shock is formulated as a free boundary. The full Euler equations are decomposed into an elliptic equation and a system of transport equations for the free boundary problem. An iteration scheme 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 the elliptic equation and the system of transport equations with the Schauder fixed point argument. The uniqueness of transonic nozzle flows is also established by employing the coordinate transformation of Euler-Lagrange type and detailed estimates of the solutions.

We show the existence of weak solutions in an elliptic region in the self-similar plane to the two-dimensional Riemann problem for the pressure-gradient system of the compressible Euler system. The two-dimensional Riemann problem we study is the interaction of two forward rarefaction waves, which are adjacent to a common vacuum that occupies a sectorial domain of 90 degrees. We assume the origin is on the boundary of the domain. In addition, the domain is open, bounded, and simply connected with a piecewise C 2, boundary. We resolve the difficulty that arises from the fact that the origin is on the boundary of the domain.

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We study the regularity of semihyperbolic patches of self-similar solutions near sonic lines to a Riemann problem for the two-dimensional (2-D) Euler system. As a result, it is verified that there exists a global solution in the semihyperbolic patch up to the sonic boundary and that the sonic boundary has C 1 -regularity. The study of the semihyperbolic patches of solutions for the Euler system was initiated by Li and Zheng [Arch. This type of solution appears in the transonic flow over an airfoil and Guderley reflection and is common in the numerical configurations of 2-D Riemann problems. Introduction. Previously, Li and Zheng [8]established the global existence of smooth solutions in the semihyperbolic patches for the isentropic Euler system, which is frequently observed in several numerical configurations of the two-dimensional (2-D) Riemann problems of the Euler system. In this paper, we improve the result of [8]; namely, we obtain the global existence of solutions up to the sonic lines, where degeneracy of hyperbolicity occurs, in the semihyperbolic region for the isentropic Euler system, and, moreover, we also verify that the sonic lines are C 1 continuous. This paper concerns sonic lines and semihyperbolic regions arising from the transonic problems of the 2-D isentropic Euler system. Recently, there have been many of works on multidimensional transonic problems using various approximate models of multidimensional gas flows. In particular, the research in [1,7,12,13,16,17] was developed under specific physical models such as infinite long nozzle problems or shock reflection problems, etc. It is verified that the problems on the sonic lines and semihyperbolic regions are not only from the above physical situations but also 2-D four-wave Riemann problems.In general, the initial value problem of the 2-D full Euler system is regarded as very difficult. One of the ways to handle this problem is to consider simple initial data. The Riemann problem is such a case. In this paper, we will consider the 2-D isentropic compressible Euler system

We establish the existence of the classical solution for the pressure-gradient equation in a nonsmooth and non-convex domain. The equation is elliptic inside the domain, becomes degenerate on the boundary, and is singular at the origin when the origin lies on the boundary. We show the solution is smooth inside the domain and continuous up to the boundary.  2004 Elsevier Inc. All rights reserved.