Conservation Laws and Shock Waves

Abstract

Conservation laws describe the conservation of some basic physical quantities of a system and they arise in all branches of science and engineering. In this chapter, we study first-order, quasi-linear, partial differential equations which become conservation laws. We discuss the fundamental role of characteristics in the study of quasi-linear equations and then solve the nonlinear, initial-value problems with both continuous and discontinuous initial data. Special attention is given to discontinuous (or weak) solutions, development of shock waves, and breaking phenomena. As we have observed, quasi-linear equations arise from integral conservation laws which may be satisfied by functions which are not differentiable, and not even continuous, but simply bounded and measurable. These functions are called weak or generalized solutions, in contrast to classical solutions, which are smooth (differentiable) functions. It is shown that the integral conservation law can be used to derive the jump condition, which allows us to determine the speed of discontinuity or shock waves. Finally, a formal definition of a shock wave is given.

The strides that have been made recently, in the theory of nonlinear partial differential equations, are as great as in the linear theory. Unlike the linear case, no wholesale liquidation of broad classes of problems has taken place; rather, it is steady progress on old fronts and on some new ones, the complete solution of some special problems, and the discovery of some brand new phenomena. The old tools—variational methods, fixed-point theorems, mapping degree, and other topological tools have been augmented by some new ones. Pre-eminent for discovering new phenomena is numerical experimentation; but it is likely that in the future numerical calculations will be parts of proofs.

Peter Lax