In this work, I will present recent work on high-frequency solutions to the Einstein vacuum equations. From a physical point of view, these solutions model high-frequency gravitational waves and describe how waves travel on a fixed background metric. As noted in 1969 by Choquet-Bruhat, a wave has an effect on the background on which it travels. This is the so called backreaction phenomenon, which is also at the heart of the Burnett conjecture (stated in 1989). This conjecture adresses the lack of compactness of the family of vacuum spacetimes, for a sufficiently weak topology allowing in the limit non-trivial contribution to the matter side of the Einstein vacuum equations. The goal of my talk is to show how one can tackle those questions from the Cauchy problem in general relativity and what are the PDE challenges encountered. More precisely, after reviewing the literature on high-frequency gravitational waves and on the Burnett conjecture, I will present my work on a toy model. I will then conclude my talk by sketching the proof of the local existence in harmonic gauge of high-frequency gravitational waves.
