Global Existence for Nonlinear Wave Equations with Multiple Speeds

The main difficulty in the nonrelativistic case is that one can only use a smaller group of commuting vector fields. In particular, since one cannot use the generators of the hyperbolic rotations, due to the different wave speeds, the earlier approach of Klainerman [9] breaks down. This is because the invariant Sobolev inequality that plays a key role in [9] does not hold if one uses a smaller collection of vector fields. The conformal approach of Christodoulou [2] also does not seem to apply to the nonrelativistic approach. Also, unlike [9], our techniques do not use Morawetz’s conformal vector field.