Hyperbolic Systems of Conservation Laws

Conservation laws are first order systems of quasilinear partial differential equations in divergence form; they express the balance laws of continuum physics for media with "elastic" response, in which internal dissipation is neglected. The absence of internal dissipation is manifested in the emergence of solutions with jump discontinuities across manifolds of codimension one, representing, in the applications, phase boundaries or propagating shock waves. The presence of discontinuities makes the analysis hard; the redeeming feature is that solutions are endowed with rich geometric structure. Indeed, the most interesting results in the area have a combined analytic-geometric flavor. The paper will survey certain aspects of the theory of hyperbolic systems of conservation laws. Any attempt to be comprehensive would fail because of space limitations. Major developments in the future will likely come from the exploration of systems in several space dimensions, which presently is terra incognita. Accordingly, the author has opted to emphasize the multidimensional setting, at the expense of the one-space dimensional case, where past and present achievements mostly lie. Many important recent results in one-space dimension will only be briefly mentioned, whereas others will not be referenced at all. Similarly, the bibliography is far from exhaustive. Finally, the exciting research program that addresses the connection between systems of conservation laws and the kinetic equations will not be discussed here as it will be surveyed in this volume by Perthame [P].