Finite speed of propagation

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Finite speed of propagation refers to the phenomenon for certain equations that information (such as support or singularities of solutions) only propagate at a bounded speed. In general, relativisitic equations such as nonlinear wave and Klein-Gordon equations enjoy finite speed of propagation, whereas highly dispersive equations such as Schrodinger equations or KdV-type equations do not (see infinite speed of propagation).

One can determine finite or infinite speed of propagation in a number of ways. One is via the dispersion relation. Another is by inspection of a fundamental solution. A third is by energy estimates, for instance by exploiting the positivity properties of the stress-energy tensor.

Relativistic equations

In relativistic equations, information only propagates at the speed of light c (typically normalized to 1) or slower; for massless equations such as the free wave equation, information in fact propagates at exactly c. Singularities also tend to propagate at exactly c.

Finite speed of propagation allows one to localize space whenever time is localized. Because of this, there is usually no distinction between periodic and non-periodic wave equations. Another application is to convert local existence results for large data to that of small data (though in sub-critical situations this is often better achieved by scaling or similar arguments). Also, the behaviour of blowup at a point is only determined by the solution in the backwards light cone from that point; thus to avoid blowup one needs to show that the solution cannot concentrate into a backwards light cone. One can also use finite speed of propagation to truncate constant-in-space solutions (which evolve by some simple ODE) to obtain localized solutions. This is often useful to demonstrate blowup for various focussing equations.

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