A proof of existence of perturbed steady transonic shocks via a free boundary problem

We prove the existence of a solution of a free boundary problem for the tra nsonic small-disturbance equation. The free boundary is the position of a t ransonic shock dividing two regions of smooth now. Assuming inviscid, irrot ational flow, as modeled by the transonic small-disturbance equation, the e quation is hyperbolic upstream where the flow is supersonic, and elliptic i n the downstream subsonic region. To study the stability of a uniform plana r transonic shock, we consider perturbation by a steady Cl + epsilon upstre am disturbance, If the upstream disturbance is small in a C-l sense, then t here is a steady solution in which the downstream flow and the transonic sh ock rue Holder-continuous perturbations of the uniform configuration. This result provides a new use of inviscid perturbation techniques to demonstrat e, in two dimensions, the stability of transonic shock waves of the type th at appear, for example, over the wing of an airplane, along an airfoil, or as bow shacks in a how with a supersonic fret-stream velocity. (C) 2000 Joh n Wiley & Sons, Inc.