# Scalar Conservation Laws and First Order Equations

• Sandro Salsa
Chapter
Part of the Universitext book series (UTX)

## Abstract

In the first part of this chapter we consider equations of the form
$${u_t} + q{(u)_x} = 0,x \in \mathbb{R},t > 0.$$
(4.1)
In general, u = u (x, t) represents the density or the concentration of a physical quantity Q and q (u) is its flux function1. Equation (4.1) constitutes a link between density and flux and expresses a (scalar) conservation law for the following reason. If we consider a control interval [x1, x2], the integral
$$\int_{{x_1}}^{{x_2}} {u(x,t)dx}$$
gives the amount of Q between x1 and x2 at time t. A conservation law states that, without sources or sinks, the rate of change of Q in the interior of [x1, x2] is determined by the net flux through the end points of the interval.

## Keywords

Weak Solution Conservation Laws Cauchy Problem Travel Wave Solution Order Equation
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.