Encyclopedia of Complexity and Systems Science

Living Edition
| Editors: Robert A. Meyers

Hyperbolic Conservation Laws

  • Alberto Bressan
Living reference work entry
DOI: https://doi.org/10.1007/978-3-642-27737-5_278-2


Conservation law

Several physical laws state that certain basic quantities such as mass, energy, or electric charge are globally conserved. A conservation law is a mathematical equation describing how the density of a conserved quantity varies in time. It is formulated as a partial differential equation having divergence form.

Flux function

The flux of a conserved quantity is a vector field describing how much of the given quantity moves across any surface at a given time.


Solutions to conservation laws often develop shocks, i.e., surfaces across which the basic physical fields are discontinuous. Knowing the two limiting values of a field on opposite sides of a shock, one can determine the speed of propagation of a shock in terms of the Rankine–Hugoniot equations.


Entropy is an additional quantity which is globally conserved for every smooth solution to a system of conservation laws. In general, however, entropies are not conserved by solutions containing shocks....


Weak Solution Space Dimension Hyperbolic System Riemann Problem Piecewise Constant Function 
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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© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Department of MathematicsPenn State UniversityUniversity ParkUSA