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How to get a conservative well-posed linear system out of thin air. Part II. Controllability and stability
Published versio
Cross-shaped and Degenerate Singularities in an Unstable Elliptic Free Boundary Problem
We investigate singular and degenerate behavior of solutions of the unstable
free boundary problem First, we construct a
solution that is not of class and whose free boundary consists of
four arcs meeting in a {\em cross}-shaped singularity. This solution is
completely unstable/repulsive from above and below which would make it hard to
get by the usual methods, and even numerics is non-trivial. We also show
existence of a degenerate solution. This answers two of the open questions in a
recent paper by R. Monneau-G.S. Weiss
Self-propagating High temperature Synthesis (SHS) in the high activation energy regime
We derive the precise limit of SHS in the high activation energy scaling
suggested by B.J. Matkowksy-G.I. Sivashinsky in 1978 and by A. Bayliss-B.J.
Matkowksy-A.P. Aldushin in 2002. In the time-increasing case the limit turns
out to be the Stefan problem for supercooled water with spatially inhomogeneous
coefficients. Although the present paper leaves open mathematical questions
concerning the convergence, our precise form of the limit problem suggest a
strikingly simple explanation for the numerically observed pulsating waves
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