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

Abstract

AbstractWe establish the existence and stability of multidimensional transonic shocks (hyperbolic‐elliptic shocks) for the Euler equations for steady compressible potential fluids in infinite cylinders. The Euler equations, consisting of the conservation law of mass and the Bernoulli law for velocity, can be written as a second order nonlinear equation of mixed elliptic‐hyperbolic type for the velocity potential. The transonic shock problem in an infinite cylinder can be formulated into the following free boundary problem: The free boundary is the location of the multidimensional transonic shock which divides two regions of C1,α flow in the infinite cylinder, and the equation is hyperbolic in the upstream region where the C1,α perturbed flow is supersonic. We develop a nonlinear approach to deal with such a free boundary problem in order to solve the transonic shock problem in unbounded domains. Our results indicate that there exists a solution of the free boundary problem such that the equation is always elliptic in the unbounded downstream region, the uniform velocity state at infinity in the downstream direction is uniquely determined by the given hyperbolic phase, and the free boundary is C1,α, provided that the hyperbolic phase is close in C1,α to a uniform flow. We further prove that, if the steady perturbation of the hyperbolic phase is C2,α, the free boundary is C2,α and stable under the steady perturbation. © 2003 Wiley Periodicals Inc.