Talk Title: Self-Similar Solutions to 2-D Riemann Problems
for Hyperbolic Conservation Laws
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Abstract:
Although multi-dimensional hyperbolic conservation laws arise in many applications
(such as aerodynamics, industrial gas processing and environmental science),
the development of the general theory of solutions
is rather challenging.
The main feature of hyperbolic conservation laws is the finite-time formation
of singularities (or shock waves) in solutions no matter how smooth the
initial data are.
This causes major difficulties even in one space dimension.
In two or more space dimensions, the added degrees of freedom
allow formation of singularities that cannot occur in
one space dimension.
This is why standard, well-developed, one-dimensional
techniques do not generalize to more than one space dimension.
In fact, even when self-similar solutions to two-dimensional
(Riemann) problems are considered, new techniques,
based on the study of mixed, hyperbolic-elliptic systems,
need to be employed to understand the structure of solutions.
This can be seen in a benchmark problem
of weak shock reflection by a wedge.
In a collaborative effort with E-H. Kim, B.L. Keyfitz and G. Lieberman
we have successfully resolved
global existence and structure of self-similar solutions
for a class of two-dimensional Riemann problems
which includes regular reflection of weak shocks modeled by the
unsteady transonic small disturbance equation.
An outline of the methods and results will be presented,
and a comparison with the related works in this field
will be given.
Generalization of the techniques to Mach and von Neumann
reflection of weak shocks will be outlined.
A proposal for the analysis of shock reflection modeled
by the isentropic Euler equations and the
nonlinear wave system  will be outlined.