Partial Differential Equations in Action pp 156-220 | Cite as

# Scalar Conservation Laws and First Order Equations

Chapter

## Abstract

In the first part of this chapter we consider equations of the form In general, gives the amount of

$$
{u_t} + q{(u)_x} = 0,x \in \mathbb{R},t > 0.
$$

(4.1)

*u*=*u*(*x*,*t*) represents the*density*or the*concentration*of a physical quantity*Q*and*q*(*u*) is its*flux function*^{1}. 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 [*x*_{1},*x*_{2}], the integral$$
\int_{{x_1}}^{{x_2}} {u(x,t)dx}
$$

*Q*between*x*_{1}and*x*_{2}at time*t*. A*conservation law*states that, without sources or sinks, the rate of change of*Q*in the interior of [*x*_{1},*x*_{2}] 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.

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## Copyright information

© Springer-Verlag Italia 2008