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Euler Equations and Related Hyperbolic Conservation Laws

#### Author: Gui-Qiang Chen

#### Title:
Euler Equations and Related Hyperbolic Conservation Laws

**Abstract**

Some aspects of recent developments in the study of the Euler
equations for compressible fluids and related hyperbolic
conservation laws are analyzed and surveyed. Basic features and
phenomena including convex entropy, symmetrization, hyperbolicity,
genuine nonlinearity, singularities, $BV$ bound, concentration, and
cavitation are exhibited. Global well-posedness for discontinuous solutions,
including the $BV$ theory and the $L^\infty$ theory, for the
one-dimensional Euler equations and related hyperbolic systems of
conservation laws is described. Some analytical approaches including
techniques, methods, and ideas, developed recently, for solving
multidimensional steady problems are presented. Some
multidimensional unsteady problems are analyzed. Connections between
entropy solutions of hyperbolic conservation laws and
divergence-measure fields, as well as the theory of
divergence-measure fields, are discussed. Some further trends and
open problems on the Euler equations and related multidimensional
conservation laws are also addressed.

This article has appeared in:

* The Handbook of Differential Equations *, **Vol. 2**, pages 1-104 (2005), Edited by C. M. Dafermos and E. Feireisl, Elsevier Science B. V.

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**Author Address**
Department of Mathematics
Northwestern University
Evanston, IL 60208-2730
gqchen@math.northwestern.edu