Discontinuous Solutions of Conservation Laws

Abstract

In this chapter we shall begin the study quasi-linear systems of the form

$$ {u_t} + f{\left( u \right)_x} = 0, $$

(15.1)

where u = (u₁,…, u _n ) ∈ Rⁿ, n ≥ 1, and (x, t) ∈ R x R₊. We assume that the vector-valued function f is C² in some open subset Ω ⊂ Rⁿ. These equations are commonly called conservation laws in analogy to the examples of such systems which arise in physics; see the examples below.