EXISTENCE AND STABILITY OF MULTIDIMENSIONAL TRANSONIC FLOWS THROUGH AN INFINITE NOZZLE OF ARBITRARY CROSS-SECTIONS GUI-QIANG CHEN AND MIKHAIL FELDMAN Abstract. We establish the existence and stability of multidimensional steady transonic flows with transonic shocks through an infinite nozzle of arbitrary cross-sections, including a slowly varying de Laval nozzle. The transonic flow is governed by the inviscid potential flow equation with supersonic upstream flow at the entrance, uniform subsonic downstream flow at the exit at infinity, and the slip boundary condition on the nozzle boundary. Our results indicate that, if the supersonic upstream flow at the entrance is sufficiently close to a uniform flow, there exists a solution that consists of a C 1,α subsonic flow in the unbounded downstream region, converging to a uniform velocity state at infinity, and a C 1,α multidimensional transonic shock dividing the subsonic flow from the supersonic upstream flow; the uniform velocity state at the exit at infinity in the downstream direction is uniquely determined by the supersonic upstream flow; and the shock is orthogonal to the nozzle boundary at every point of their intersection. In order to construct such a transonic flow, we reformulate the multidimensional transonic nozzle problem into a free boundary problem for the subsonic phase, in which the equation is elliptic and the free boundary is a transonic shock. The free boundary conditions are determined by the Rankine-Hugoniot conditions along the shock. We further develop a nonlinear iteration approach and employ its advantages to deal with such a free boundary problem in the unbounded domain. We also prove that the transonic flow with a transonic shock is unique and stable with respect to the nozzle boundary and the smooth supersonic upstream flow at the entrance. 1. Introduction We are concerned with the existence and stability of multidimensional steady transonic flows with transonic shocks through multidimensional infinite nozzles of arbitrary cross-sections. Such problems naturally arise in many physical situations, especially in the de Laval nozzles which have widely been used in the design of steam turbines and modern rocket engines (see Courant- Friedrichs [11], Whitham [47], and the references cited therein). Since the nozzles in applications are usually much longer with respect to their cross-sections, the problem is often formulated mathematically as an infinite nozzle problem. Correspondingly, such a multidimensional infinite nozzle problem has extensively been studied experimentally, computationally, and asymptotically (see [11, 15, 18, 21, 22, 23, 47] and the references cited therein). The stability issue for transonic gas flow through a nozzle for the one-dimensional model was analyzed in [40, 41]. The existence and stability of multidimensional steady transonic flows through multidimensional infinite nozzles has remained open; see [5, 11, 12, 43, 47]. In this paper, we focus on the infinity nozzle to establish the existence and stability of mul- tidimensional transonic flows with supersonic upstream flows at the entrance, uniform subsonic downstream flows at the exit at infinity, and the slip boundary condition on the nozzle bound- ary. The potential flow equation for the velocity potential ϕ :Ω ⊂ R n → R is a second-order Date : November 1, 2005. 1991 Mathematics Subject Classification. 35M10,35J65,35R35,76H05,76L05,35B45. Key words and phrases. Elliptic-hyperbolic, nonlinear equations, second order, mixed type, multidimensional transonic shocks, free-boundary problems, unbounded domains, infinite nozzles, cross-sections, existence, unique- ness, stability, Euler equations, inviscid potential flow. 1