The initial value problem for a scalar conservation law

Abstract

Many physical applications involve one or several conservation laws. One typical example is the equation of continuity for Kynch’s theory of sedimentation. Since the pioneering works of Lax (1957) and Oleinik (1957) on conservation laws, the field has been greatly developed as we show in this and the following chapters. We begin with some general properties of conservation laws. Let ƒ : [a ₁,a ₂] → ℝ, ƒ ∈ C ³ be a nonlinear function of φ (in particular, f is Lipschitz continuous then), and let Ω = {(z, t)|;z ‘ ℝ, t > 0}. We consider the quasilinear hyperbolic equation

$$\frac{{\partial \varphi }}{{\partial t}} + \frac{{\partial f(\varphi )}}{{\partial z}} = 0\,in\,\Omega $$

((4.1a))

and the initial condition

$$ \varphi (z,0) = {\varphi _I}(z)\,for\,z\, \in \,$$

((4.1b))

.