Concentration in Solutions to Hyperbolic Conservation Laws
Authors: Gui-Qiang Chen
Title:
Concentration in Solutions to Hyperbolic Conservation Laws
Abstract
The phenomenon of concentration in solutions to hyperbolic systems
of conservation laws and related recent analytical techniques
for studying such solutions are discussed.
In particular, for the Euler equations for both isentropic and
nonisentropic fluids, the phenomenon of concentration in solutions
is fundamental as the pressure vanishes,
which occurs not only in the multidimensional situations, but also
even naturally in the one-dimensional case.
From the point of view of hyperbolic conservation laws,
since the limit system loses hyperbolicity,
the phenomenon of concentration in the process of
vanishing pressure limit can be regarded as a phenomenon
of resonance among the characteristic fields.
This phenomenon indicates that the flux-function limit can be very
singular: the limit functions of solutions are no longer in the spaces of
functions, $BV$ or $L^\infty$, yet the space of Radon measures,
for which the divergences of certain entropy and entropy flux fields
are also Radon measures, is a natural space
to deal with such a limit in general.
In this regard, a theory of divergence-measure
fields developed recently is also presented.
This theory especially includes normal traces, a generalized
Gauss-Green theorem, and product rules, among others.
Some applications of this theory to solving various nonlinear problems
in conservation laws and related areas are also discussed.
This article has appeared in:
Contemporary Mathematics
vol. 327, pages 41-60, 2003.
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