Related papers: Energy estimate for initial data on a characterist…
We construct a black hole initial data for the Einstein equations with prescribed scalar curvature, or more precisely a piece of initial data contained inside the black hole. The constraints translate into a parabolic equation, with radius…
This paper initiates the study of the Einstein equation on homogeneous supermanifolds. First, we produce explicit curvature formulas for graded Riemannian metrics on these spaces. Next, we present a construction of homogeneous…
We give a short proof of the existence of a small piece of null infinity for $(3+1)$-dimensional spacetimes evolving from asymptotically flat initial data as solutions of the Einstein vacuum equations. We introduce a modification of the…
We present a new scheme for constructing initial data for the Einstein field equations using the conformal thin-sandwich formulation that does not assume conformal flatness or approximate Killing vectors. This includes a method for…
The Cauchy+characteristic matching (CCM) problem for the scalar wave equation is investigated in the background geometry of a Schwarzschild black hole. Previously reported work developed the CCM framework for the coupled…
This article is a guide to theorems on existence and global dynamics of solutions of the Einstein equations. It draws attention to open questions in the field. The local-in-time Cauchy problem, which is relatively well understood, is…
We prove energy estimates for solutions to a tensorial system of coupled non-linear wave equations, in a way that is suitable to deal with the structure of the non-linearity that arises from the Einstein-Yang-Mills system in the Lorenz…
We propose a numerical method to solve an inverse source problem of computing the initial condition of hyperbolic equations from the measurements of Cauchy data. This problem arises in thermo- and photo- acoustic tomography in a bounded…
We study the global existence of Einstein-Maxwell(EM) equations on $\mathbb{R}^4$. We use the method, which relies on wave and Lorentzian gauge conditions, to obtain some exquisite estimates. Our main conclusion is that if the initial data…
We develop an efficient and convergent numerical method for solving the inverse problem of determining the potential of nonlinear hyperbolic equations from lateral Cauchy data. In our numerical method we construct a sequence of linear…
We consider the Cauchy problem for wave equations with localized damping in ${\bf R}^{2}$. The damping is effective only near spatial infinity. We obtain fast energy decay estimate such that $O(t^{-2}\log t)$ as $t \to \infty$. Unlike the…
A class of gauges for the Einstein vacuum equations is introduced, along with three symmetric hyperbolic systems. The first implies the local realizability of the gauge. The second is the dynamical subset of the field equations. The third…
This thesis describes the application of numerical techniques to solve Einstein's field equations in three distinct cases. First we present the first long-term stable second order convergent Cauchy characteristic matching code in…
This article is a guide to theorems on existence and global dynamics of solutions of the Einstein equations. It draws attention to open questions in the field. The local in time Cauchy problem, which is relatively well understood, is…
In certain models of conformal gravity, the propagation of gravitational waves is governed by a fourth order scalar partial differential equation. We study the initial value problem for a generalization of this equation, and derive a…
We establish both global existence and decay properties for solutions with small data for a general class of coupled system of tensorial quasilinear hyperbolic wave equations in three space dimensions, that covers the dynamical Einstein…
This is the second paper of a two part work that establishes a definitive quantitative nonlinear scattering theory for asymptotically de Sitter vacuum solutions $(M,g)$ in $(n+1)$ dimensions with $n\geq4$ even, which are determined by small…
A method of solving the eikonal equation, in either flat or curved space-times, with arbitrary Cauchy data, is extended to the case of data given on a characteristic surface. We find a beautiful relationship between the Cauchy and…
We study the local existence of strong solutions for the cubic nonlinear wave equation with data in $H^s(M)$, $s<1/2$, where $M$ is a three dimensional compact riemannian manifold. This problem is supercritical and can be shown to be…
In this paper we study the spherically symmetric characteristic initial data problem for the Einstein-Maxwell-scalar field system with a positive cosmological constant in the interior of a black hole, assuming an exponential Price law along…