## Convexity of Self-Similar Transonic Shocks and Free Boundaries for Potential Flow. (arXiv:1803.02431v1 [math.AP])

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
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