The Cauchy Problem

Abstract

The theory of the Cauchy problem for hyperbolic conservation laws is confronted with two major challenges. First, classical solutions, starting out from smooth initial values, spontaneously develop discontinuities; hence in general only weak solutions may exist in the large. Next, weak solutions to the Cauchy problem fail to be unique. One does not have to dig too deep in order to encounter these difficulties. As shown in Sections 4.2 and 4.4, they arise even at the level of the simplest nonlinear hyperbolic conservation laws, in one or several space dimensions.

The Cauchy problem for weak solutions will be formulated in Section 4.3. To overcome the obstacle of nonuniqueness, restrictions need to be imposed that will weed out unstable, physically irrelevant, or otherwise undesirable solutions, in hope of singling out a unique admissible solution. Two admissibility criteria will be introduced in this chapter: the requirement that admissible solutions satisfy a designated entropy inequality; and the principle that admissible solutions should be limits of families of solutions to systems containing diffusive terms, as the diffusion asymptotically vanishes. A preliminary comparison of these criteria will be conducted.

The chapter will close with the formulation of the initial-boundary value problem for hyperbolic conservation laws.