Part of the Applied Mathematical Sciences book series (AMS, volume 152)


Hyperbolic conservation laws are partial differential equations of the form
$$\displaystyle\frac{\partial u}{\partial t}+\nabla\cdot f(u)=0.$$
If we write \(f=\left(f_{1},\dots,f_{m}\right)\), \(x=\left(x_{1},x_{2},\dots,x_{m}\right)\in\mathbb{R}^{m}\), and introduce initial data u 0 at t = 0, the Cauchy problem for hyperbolic conservation laws reads
$$\displaystyle\frac{\partial u(x,t)}{\partial t}+\sum_{j=1}^{m}\frac{\partial}{\partial x_{j}}f_{j}\left(u(x,t)\right)=0,\quad u|_{t=0}=u_{0}.$$
In applications, t normally denotes the time variable, while x describes the spatial variation in m space dimensions. The unknown function u (as well as each f j ) can be a vector, in which case we say that we have a system of equations, or u and each f j can be a scalar. This book covers the theory of scalar conservation laws in several space dimensions as well as the theory of systems of hyperbolic conservation laws in one space dimension. In the present chapter we study the one-dimensional scalar case to highlight some of the fundamental issues in the theory of conservation laws.


Weak Solution Classical Solution Riemann Problem Entropy Solution Quasilinear Equation 
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© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Department of Mathematical SciencesNorwegian University of Science and TechnologyTrondheimNorway
  2. 2.Department of MathematicsUniversity of OsloOsloNorway

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