Speaker: Hans Lindblad (Johns Hopkins University)

Title:  Global Existence and Scattering for Einstein’s equations and related equations satisfying the weak null condition

Abstract: Einstein’s equations in harmonic or wave coordinates are a system of nonlinear wave equations for a Lorentzian metric, that in addition satisfy the preserved wave coordinate condition. Christodoulou-Klainerman proved global existence for Einstein vacuum equations for small asymptotically flat initial data. Their proof avoids using coordinates since it was believed the metric in harmonic coordinates would blow up for large times. John had noticed that solutions to some nonlinear wave equations blow up for small data, whereas Klainerman came up with the ‘null condition’, that guaranteed global existence for small data. However Einstein’s equations do not satisfy the null condition. Hormander introduced a simplified asymptotic system by neglecting angular derivatives which we expect decay faster due to the rotational invariance, and used it to study blowup. I showed that the asymptotic system corresponding to the quasilinear part of Einstein’s equations does not blow up and gave an example of a nonlinear equation of this form that has global solutions even though it does not satisfy the null condition. Together with Rodnianski we introduced the ‘weak null condition’ requiring that the corresponding asymptotic system have global solutions and we showed that Einstein’s equations in wave coordinates satisfy the weak null condition and we proved global existence for this system. Our method reduced the proof to a fraction and has now been used to prove global existence also with matter fields.

Recently I derived precise asymptotics for the metric which involves logarithmic corrections to the radiation field of solutions of linear wave equations. We are further imposing these asymptotics at infinity and solve the equations backwards to obtain global solutions with given data at infinity.