Convexity of Self-Similar Transonic Shocks and Free Boundaries for Potential Flow

Gui-Qiang G. Chen, Mikhail Feldman, Wei Xiang

Published 2018-03-06Version 1

We are concerned with the convexity of transonic shocks in two-dimensional self-similar coordinates for compressible fluid flows, which not only arises in continuum physics but also is fundamental in the mathematical theory of multidimensional conservation laws. We first develop a general framework under which self-similar transonic shock waves, as free boundaries, are proved to be uniformly convex for potential flow and then apply this framework to the complete proof of the uniform convexity of transonic shocks in the two longstanding fundamental shock problems -- the shock reflection-diffraction problem by wedges and the Prandtl-Meyer reflection problem for supersonic flows past solid ramps. To achieve this, we develop a nonlinear approach to explore the detailed nonlocal behavior of the solution on the boundary to prove the uniform convexity of the transonic shock. This approach and related techniques, developed here, will also be useful for other related nonlinear problems involving similar analytical difficulties.