The term "hyperbolic systems of conservation laws" is used for a large class of systems of partial differential equations which arise in several areas of continuum physics. Each equation of the system is a "balance law". Continuum physics studies the behavior of continuous media and their so-called "extensive quantities" (like, for instance, mass, momentum and energy). These are monitored through the fields of their densities. A balance law stipulates that the time rate of change in the amount of an extensive quantity stored inside any subdomain of the body is balanced by its flux through the boundary of the subdomain.

Therefore, balance laws have the following form

q_t + div_x F = 0

where the flux F is typically expressed as a function of q and of the other quantities describing the system.


Perhaps the most famous hyperbolic system of conservation laws is the "compressible Euler", which describes the dynamic of certain gases. Solutions of hyperbolic conservation laws may be visualized as propagating waves. When the system is nonlinear jump discontinuities, known as shocks, typically arise in finite time, generated by compression waves whose profile gets progressively steeper and eventually breaks down. Therefore the theory of solutions for these systems of partial differential equations must inevitably deal with weak solutions. However, in the realm of weak solutions uniqueness is lost and one needs extra admissibility conditions to single out "physically admissible" ones.