Related papers: Ill-posedness in the Einstein equations
In a recent paper the first coauthor presented a new parabolic extension (PADM) of the standard 3+1 Arnowitt, Deser, Misner formulation of the equations of general relativity. By parabolizing first-order ADM in a certain way, the PADM…
We study the local in time well-posedness of the initial boundary value problem (IBVP) for the vacuum Einstein equations in general relativity with geometric boundary conditions. For conformal-mean curvature boundary conditions, consisting…
This paper develops a method for solving Einstein's equation numerically on multi-cube representations of manifolds with arbitrary spatial topologies. This method is designed to provide a set of flexible, easy to use computational…
We study regularization of ill-posed equations involving multiplication operators when the multiplier function is positive almost everywhere and zero is an accumulation point of the range of this function. Such equations naturally arise…
In the literature on singular perturbation (Lavrentiev regularization) for the stable approximate solution of operator equations with monotone operators in the Hilbert space the phenomena of conditional stability and local well-posedness…
Recently one and two-parameter deformed Einstein equations have been studied for extremal quantum black holes which have been proposed to obey deformed statistics by Strominger. In this study, we give a deeper insight to the deformed…
An almost Einstein manifold satisfies equations which are a slight weakening of the Einstein equations; Einstein metrics, Poincare-Einstein metrics, and compactifications of certain Ricci-flat asymptotically locally Euclidean structures are…
Instabilities in finite difference codes due to the singularity of spherical coordinates at the center are studied. In typical Numerical Relativity applications, standard regularization techniques by themselves do not ensure long term…
The demand to obtain answers to highly complex problems within strong-field gravity has been met with significant progress in the numerical solution of Einstein's equations - along with some spectacular results - in various setups. We…
The Einstein equations have proven surprisingly difficult to solve numerically. A standard diagnostic of the problems which plague the field is the failure of computational schemes to satisfy the constraints, which are known to be…
We present a general approach to the analysis of gauge stability of 3+1 formulations of General Relativity (GR). Evolution of coordinate perturbations and the corresponding perturbations of lapse and shift can be described by a system of…
The conformal formulation of the Einstein constraint equations has been studied intensively since the modern version of the conformal method was first pub- lished in the early 1970s. Proofs of existence and uniqueness of solutions were…
We extend the notion of numerical stability of finite difference approximations to include hyperbolic systems that are first order in time and second order in space, such as those that appear in Numerical Relativity. By analyzing the symbol…
We study the local well-posedness of the initial value problem for cubic Horndeski theories. Three different strongly hyperbolic modifications of the ADM formulation of the Einstein equations are extended to cubic Horndeski theories in the…
Consider an operator equation (*) $B(u)-f=0$ in a real Hilbert space. Let us call this equation ill-posed if the operator $B'(u)$ is not boundedly invertible, and well-posed otherwise. The DSM (dynamical systems method) for solving equation…
We show that the canonical formulation of the semiclassical Einstein equation, where the matter terms in the constraints are replaced by expectation values of the corresponding operators in quantum states, is inconsistent due to the…
The application of numerical relativity to cosmological spacetimes is providing new insights into the behavior of Einstein's equations, beyond common approximations. In order for simulations to be performed as accurately and efficiently as…
The Hausdorf moment problem (HMP) over the unit interval in an $L^2$-setting is a classical example of an ill-posed inverse problem. Since various applications can be rewritten in terms of the HMP, it has gathered significant attention in…
We construct solutions with prescribed asymptotics to the Einstein constraint equations using a cut-off technique. Moreover, we give various examples of vacuum asymptotically flat manifolds whose center of mass and angular momentum are…
Professor Pierre Sabatier contributed much to the study of inverse problems in theory and practice. Two of these contributions were a focus on theory that actually supports practice, and the identification of well-posed aspects of inverse…