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 ∞ 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.
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