Elasticity

Equations arising from modeling elastic media in physics are typically generalisations of wave equations in which different components of the system may have different speeds of propagation; furthermore, the dispersion relation may not be isotropic, and thus the speed of propagation may vary with the direction of propagation.

Two-speed model

A particularly simple model for elasticity arises from a two-speed wave equation system of two fields u and v, with v propagating slower than u, e.g.

$\Box u = F(U, DU), ~\Box_s v = G(U, DU)$

where U = (u,v) and $\Box_s = s^2 \Delta - \partial_t^2$ for some 0 < s < 1. This case occurs physically when u propagates at the speed of light and v propagates at some slower speed. In this case the null forms are not as useful, however the estimates tend to be more favourable (if the non-linearities F,G are "off-diagonal") since the light cone for u is always transverse to the light cone for v. One can of course generalize this to consider multiple speed (nonrelativistic) wave equations.

Examples of two-speed models include