Hyperbolic Systems of Conservation Laws

The study of systems of quasilinear hyperbolic equations that result from the balance laws of continuum physics was initiated more than a century ago yet, despite considerable progress in recent years, most of the fundamental problems in the analytical theory remain unsolved. From the outset the student of the subject encounters obstacles such as the nonexistence of globally defined smooth solutions, as a reflection of the physical phenomenon of breaking of waves and development of shocks, and the nonuniqueness of (weak) solutions in the class of discontinuous functions. Furthermore, no strong a priori estimates have yet been discovered for systems of more than one equation so that the subject is not amenable to the functional analytic techniques that have swept through the theory of partial differential equations of other types. It is only now that a functional analytic treatment of these systems is emerging, based on the delicate theory of compensated compactness which relies upon weak a priori estimates. The reader may find an account of these interesting recent developments in the article by L. Tartar in this volume.

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