High-frequency limits and null dust shell solutions in general relativity

Understand

Consider the characteristic initial value problem for the Einstein vacuum equations without any symmetry assumptions.

• Impose a sequence of data on two intersecting null hypersurfaces, each of which is foliated by spacelike $2$-spheres.
• Assume that the sequence of data is such that the derivatives of the metrics along null directions are only uniformly bounded in $L^2$ but the derivatives of the metrics along the directions tangential to the $2$-spheres obey higher regularity bounds uniformly.
• By the results in [J.