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 transonic small-disturbance equation. The free boundary is the position of a transonic shock dividing two regions of smooth flow. Assuming inviscid, irrotational flow, as modeled by the transonic small-disturbance equation, the equation is hyperbolic upstream where the flow is supersonic, and elliptic in the downstream subsonic region. To study the stability of a uniform planar transonic shock, we consider perturbation by a steady C1+ϵ upstream disturbance. If the upstream disturbance is small in a C1 sense, then there is a steady solution in which the downstream flow and the transonic shock are Hölder-continuous perturbations of the uniform configuration. This result provides a new use of inviscid perturbation techniques to demonstrate, in two dimensions, the stability of transonic shock waves of the type that appear, for example, over the wing of an airplane, along an airfoil, or as bow shocks in a flow with a supersonic free-stream velocity.