Introduction

Abstract

Abstract A single conservation law in one space dimension is a first-order partial differential equation of the formwhere a = f is the derivative of f. For smooth solutions, the two equations (l.l) and (1.3) are entirely equivalent. If u has a jump, however, the left hand side of (1.3) will contain the product of a discontinuous function a(u) with the distributional derivative ux, which in this case contains a Dirac mass at the point of the jump. In general, such a product is not w'ell defined. Hence (1.3) is meaningful only within a class of continuous functions. On the other hand, working with the equation in divergence form (1.1) allows us to consider discontinuous solutions as well, interpreted in a distributional sense.