Scalar Conservation Laws and First Order Equations

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


In the first part of this chapter we consider equations of the form
$$ {u_t} + q{(u)_x} = 0,x \in \mathbb{R},t > 0. $$
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.


Weak Solution Conservation Laws Cauchy Problem Travel Wave Solution Order Equation 
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Copyright information

© Springer-Verlag Italia 2008

Authors and Affiliations

  • Sandro Salsa
    • 1
  1. 1.Dipartimento di MatematicaPolitecnico di MilanoItaly

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