Conservation Laws with Discontinuous Flux Functions

Abstract

In one spatial dimension, a conservation law with a space-dependent flux can be written

$$\displaystyle u_{t}+f(x,u)_{x}=0,\quad x\in\mathbb{R},\quad t> 0.$$

(8.1)

Since the interpretation of f is the flux of u at the point x, there are many applications where the flux depends on the location. We give some simple examples that are modeled by such conservation laws.

Of course it is happening inside your head, Harry,but why on earth should that mean it is not real? — Albus Dumbledore, in Harry Potter and the Deathly Hallows

Of course it is happening inside your head, Harry,but why on earth should that mean it is not real?

— Albus Dumbledore, in Harry Potter and the Deathly Hallows