Introduction

Abstract

Hyperbolic conservation laws are partial differential equations of the form

$$\displaystyle\frac{\partial u}{\partial t}+\nabla\cdot f(u)=0.$$

If we write $f=\left(f_{1},\dots,f_{m}\right)$, $x=\left(x_{1},x_{2},\dots,x_{m}\right)\in\mathbb{R}^{m}$, and introduce initial data u ₀ at t = 0, the Cauchy problem for hyperbolic conservation laws reads

$$\displaystyle\frac{\partial u(x,t)}{\partial t}+\sum_{j=1}^{m}\frac{\partial}{\partial x_{j}}f_{j}\left(u(x,t)\right)=0,\quad u|_{t=0}=u_{0}.$$

(1.1)

In applications, t normally denotes the time variable, while x describes the spatial variation in m space dimensions. The unknown function u (as well as each f _j ) can be a vector, in which case we say that we have a system of equations, or u and each f _j can be a scalar. This book covers the theory of scalar conservation laws in several space dimensions as well as the theory of systems of hyperbolic conservation laws in one space dimension. In the present chapter we study the one-dimensional scalar case to highlight some of the fundamental issues in the theory of conservation laws.

I have no objection to the use of the term ‘‘Burgers equation’’ for the nonlinear heat equation(provided it is not written ‘‘Burger’s equation’’). — Letter from Burgers to Batchelor (1968)

I have no objection to the use of the term ‘‘Burgers’ equation’’ for the nonlinear heat equation(provided it is not written ‘‘Burger’s equation’’).

— Letter from Burgers to Batchelor (1968)