Hyperbolic conservation laws, degenerate elliptic equations and free
boundary problems
Barbara Keyfitz
Fields Institute and University of Houston
Abstract: This is a report on work with Suncica Canic and Eun
Heui Kim on solving degenerate elliptic problems that arise
from self-similar reductions of hyperbolic conservation laws
in two space dimensions and time. For a class of model equations,
we obtain free boundary problems for quasilinear degenerate
elliptic equations. The degeneracy results from the sonic line,
and the free boundary from the influence of the subsonic region
on the position of a shock wave.
Since the defining conditions at the free boundary are of an
unusual type, the modern approach to free boundaries does not
seem to work for these quasilinear systems, and we have developed
a classical approach based on fixed point theorems in weighted
Holder spaces. The work of Gary Lieberman on oblique derivative
and mixed boundary conditions has been very useful to us. We
have also developed modest extensions of this theory to handle
a generic situation which occurs with shock reflection problems:
the derivative boundary condition fails to be oblique at the
foot of a Mach stem and in other common situations.