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, not even continuous, 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 so called jump condition which allows determining the speed of discontinuity or shock waves. Finally, a formal definition of a shock wave is given.