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Barbara Keyfitz : Regular Reflection of Weak Shocks

In joint work, Suncica Canic, Eun Heui Kim and I have recently proved the existence of a local solution to the regular reflection problem in the unsteady transonic small disturbance (UTSD) model for shock reflection by a wedge. There are two kinds of regular reflection, weak and strong, which are distinguished by whether the state immediately behind the reflected shock is subsonic (strong) or supersonic and constant, becoming subsonic further downstream (weak). In the more complicated case of weak regular reflection, the equation, in self-similar coordinates, is degenerate at the sonic line. The reflected shock becomes transonic and begins to curve there; its position is the solution to a free boundary problem for the degenerate equation.

We combine techniques which have been developed for solving degenerate elliptic equations arising in self-similar reductions of hyperbolic conservation laws with an approach to solving free boundary problems of the type that arise from Rankine-Hugoniot relations. Although our construction is limited to a finite part of the unbounded subsonic region, it suggests that this approach has the potential to solve a variety of problems in weak shock reflection.

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