355 research outputs found
A Generalized Representation Formula for Systems of Tensor Wave Equations
In this paper, we generalize the Kirchhoff-Sobolev parametrix of Klainerman
and Rodnianski to systems of tensor wave equations with additional first-order
terms. We also present a different derivation, which better highlights that
such representation formulas are supported entirely on past null cones. This
generalization is a key component for extending Klainerman and Rodnianski's
breakdown criterion result for Einstein-vacuum spacetimes to Einstein-Maxwell
and Einstein-Yang-Mills spacetimes.Comment: 29 page
On emerging scarred surfaces for the Einstein vacuum equations
This is a follow up on our previous work in which we have presented a
modified, simpler version of the remarkable recent result of Christodoulou on
the formation of trapped surfaces. In this paper we prove two related results.
First we extend the semi-global existence result, which was at the heart of our
previous work, to an optimal range. We then use it to establish the formation
of surfaces with multiple pre-scarred angular components
On the uniqueness of solutions to the Gross-Pitaevskii hierarchy
We give a new proof of uniqueness of solutions to the Gross-Pitaevskii
hierarchy, first established by Erdos, Schlein and Yau, in a different space,
based on space-time estimates
On Breakdown Criteria for Nonvacuum Einstein Equations
The recent "breakdown criterion" result of S. Klainerman and I. Rodnianski
stated roughly that an Einstein-vacuum spacetime, given as a CMC foliation, can
be further extended in time if the second fundamental form and the derivative
of the lapse of the foliation are uniformly bounded. This theorem and its proof
were extended to Einstein-scalar and Einstein-Maxwell spacetimes in the
author's Ph.D. thesis. In this paper, we state the main results of the thesis,
and we summarize and discuss their proofs. In particular, we will discuss the
various issues resulting from nontrivial Ricci curvature and the coupling
between the Einstein and the field equations.Comment: 62 pages This version: corrected minor typos, expanded Section 6
(geometry of null cones
- …