Global existence for a system of multiple-speed wave equations violating the null condition

TL;DR

This paper proves the global existence of solutions for a system of semilinear wave equations with multiple speeds in three dimensions, despite violating the null condition, by employing advanced energy and weighted estimates.

Contribution

It extends previous results by establishing global solutions for systems with multiple speeds and null condition violations using novel energy methods.

Findings

Global solutions exist for small smooth data

The method handles multiple speeds without Lorentz boosts

Weighted energy estimates are crucial for the proof

Abstract

We discuss the Cauchy problem for a system of semilinear wave equations in three space dimensions with multiple wave speeds. Though our system does not satisfy the standard null condition, we show that it admits a unique global solution for any small and smooth data. This generalizes a preceding result due to Pusateri and Shatah. The proof is carried out by the energy method involving a collection of generalized derivatives. The multiple wave speeds disable the use of the Lorentz boost operators, and our proof therefore relies upon the version of Klainerman and Sideris. Due to the presence of nonlinear terms violating the standard null condition, some of components of the solution may have a weaker decay as $t\to\infty$, which makes it difficult even to establish a mildly growing (in time) bound for the high energy estimate. We overcome this difficulty by relying upon the ghost weight energy estimate of Alinhac and the Keel-Smith-Sogge type $L^2$ weighted space-time estimate for derivatives.