A Weak Null Structure for Multiple Speed Systems of Wave Equations

We consider the Cauchy problem in three spatial dimensions for a system of semilinear wave equations with two wave speeds. Hidano, Yokoyama, and Zha established a weak null condition that can be placed on the nonlinearities of a system of wave equations with multiple speeds to ensure its solutions exist globally in time. In this thesis, we present a new weak null structure with fewer conditions and demonstrate that it is sufficient for global existence. To do so, we will use a combination of the $r^p$-weighted energy estimate used by Dafermos and Rodnianski and the ghost weight estimate of Alinhac. We also introduce a new local energy estimate to improve our control of the solutions. In order to close the argument, we rely on the spacetime Klainerman-Sobolev estimates of Metcalfe, Tataru, and Tohaneanu.