On a System of Weakly Null Semilinear Equations

We develop a new method for addressing certain weakly null systems of wave equations. This approach does not rely on Lorentz invariance nor on the use of null foliations, both of which restrict applications to, e.g., multiple speed systems. The proof uses a class of space-time Klainerman-Sobolev estimates of the first author, Tataru, and Tohaneanu, which pair nicely with local energy estimates that combine the $r^p$-weighted method of Dafermos and Rodnianski with the ghost weight method of Alinhac. We further refine the standard local energy estimate with a modification of the ${\mathrm{\partial }}_t-{\mathrm{\partial }}_r$ portion of the multiplier.