On a class of global characteristic problems for the Einstein vacuum equations with small initial data

2010 ◽  
Vol 51 (10) ◽  
pp. 102503 ◽  
Author(s):  
Giulio Caciotta ◽  
Francesco Nicolò
Keyword(s):  
Initial Data ◽  
Small Initial Data ◽  
Einstein Vacuum Equations ◽  
Einstein Vacuum

scholarly journals GLOBAL CHARACTERISTIC PROBLEM FOR EINSTEIN VACUUM EQUATIONS WITH SMALL INITIAL DATA: (I) THE INITIAL DATA CONSTRAINTS

2005 ◽  
Vol 02 (01) ◽  
pp. 201-277 ◽  
Author(s):  
GIULIO CACIOTTA ◽  
FRANCESCO NICOLÒ
Keyword(s):  
Initial Data ◽  
Existence Result ◽  
Subsequent Paper ◽  
Small Initial Data ◽  
Asymptotic Decay ◽  
Characteristic Problem ◽  
Einstein Vacuum Equations ◽  
Einstein Vacuum ◽  
Data Constraints ◽  
Global Existence Result

We show how to prescribe the initial data of a characteristic problem satisfying the constraints, the smallness, the regularity and the asymptotic decay suitable to prove a global existence result. In this paper, the first of two, we show in detail the construction of the initial data and give a sketch of the existence result. This proof, which mimicks the analogous one for the non-characteristic problem in [19], will be the content of a subsequent paper.

scholarly journals CURVATURE BLOW UP ON A DENSE SUBSET OF THE SINGULARITY IN T3-GOWDY

2005 ◽  
Vol 02 (02) ◽  
pp. 547-564 ◽  
Author(s):  
HANS RINGSTRÖM
Keyword(s):  
Initial Data ◽  
Blow Up ◽  
Dense Subset ◽  
Cosmic Censorship ◽  
Einstein Vacuum Equations ◽  
Strong Cosmic Censorship ◽  
Einstein Vacuum ◽  
Original Argument ◽  
Gowdy Spacetimes ◽  
Generic Set

This paper is concerned with the Einstein vacuum equations under the additional assumption of T3-Gowdy symmetry. We prove that there is a generic set of initial data such that the corresponding solutions exhibit curvature blow up on a dense subset of the singularity. By generic, we mean a countable intersection of open sets (i.e. a Gδ set) which is also dense. Furthermore, the set of initial data is given the C∞ topology. This result was presented at a conference in Miami 2004. Recently, we have obtained a stronger result, but the argument to prove it is different and much longer. Therefore, we here wish to present the original argument. Finally, combining the results presented here with a paper by Chruściel and Lake, one obtains strong cosmic censorship for T3-Gowdy spacetimes.

Remarks on nonlinear elastic waves with null conditions

2020 ◽  
Vol 26 ◽  
pp. 121
Author(s):  
Dongbing Zha ◽  
Weimin Peng
Keyword(s):  
Initial Data ◽  
Global Existence ◽  
Elastic Waves ◽  
Wave Equations ◽  
Alternative Proof ◽  
Classical Solutions ◽  
Small Initial Data ◽  
Hyperelastic Materials ◽  
Variational Structure ◽  
Nonlinear Elastic

For the Cauchy problem of nonlinear elastic wave equations for 3D isotropic, homogeneous and hyperelastic materials with null conditions, global existence of classical solutions with small initial data was proved in R. Agemi (Invent. Math. 142 (2000) 225–250) and T. C. Sideris (Ann. Math. 151 (2000) 849–874) independently. In this paper, we will give some remarks and an alternative proof for it. First, we give the explicit variational structure of nonlinear elastic waves. Thus we can identify whether materials satisfy the null condition by checking the stored energy function directly. Furthermore, by some careful analyses on the nonlinear structure, we show that the Helmholtz projection, which is usually considered to be ill-suited for nonlinear analysis, can be in fact used to show the global existence result. We also improve the amount of Sobolev regularity of initial data, which seems optimal in the framework of classical solutions.

The Kerr solution

2018 ◽  
Author(s):  
Nathalie Deruelle ◽  
Jean-Philippe Uzan
Keyword(s):  
Kerr Black Hole ◽  
Geodesic Equation ◽  
Einstein Equations ◽  
Kerr Metric ◽  
Kerr Solution ◽  
Axially Symmetric ◽  
Observable Universe ◽  
Einstein Vacuum Equations ◽  
Einstein Vacuum ◽  
The Many

This chapter covers the Kerr metric, which is an exact solution of the Einstein vacuum equations. The Kerr metric provides a good approximation of the spacetime near each of the many rotating black holes in the observable universe. This chapter shows that the Einstein equations are nonlinear. However, there exists a class of metrics which linearize them. It demonstrates the Kerr–Schild metrics, before arriving at the Kerr solution in the Kerr–Schild metrics. Since the Kerr solution is stationary and axially symmetric, this chapter shows that the geodesic equation possesses two first integrals. Finally, the chapter turns to the Kerr black hole, as well as its curvature singularity, horizons, static limit, and maximal extension.

scholarly journals Global classical solutions to the elastodynamic equations with damping

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Mengmeng Liu ◽  
Xueyun Lin
Keyword(s):  
Initial Data ◽  
Periodic Boundary ◽  
Energy Estimate ◽  
Periodic Boundary Conditions ◽  
Deformation Tensor ◽  
Classical Solutions ◽  
Small Initial Data ◽  
Weighted Energy ◽  
Global Classical Solutions ◽  
Weighted Energy Estimate

AbstractIn this paper, we show the global existence of classical solutions to the incompressible elastodynamics equations with a damping mechanism on the stress tensor in dimension three for sufficiently small initial data on periodic boxes, that is, with periodic boundary conditions. The approach is based on a time-weighted energy estimate, under the assumptions that the initial deformation tensor is a small perturbation around an equilibrium state and the initial data have some symmetry.

Singularities of solutions for first order quasilinear hyperbolic systems

1983 ◽  
Vol 94 (1-2) ◽  
pp. 137-147 ◽  
Author(s):  
Lee Da-tsin(Li Ta-tsien) ◽  
Shi Jia-hong
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Hyperbolic Systems ◽  
Smooth Solutions ◽  
Small Initial Data ◽  
Quasilinear Hyperbolic Systems ◽  
First Order ◽  
Global Smooth Solutions ◽  
The Cauchy Problem

SynopsisIn this paper, the existence of global smooth solutions and the formation of singularities of solutions for strictly hyperbolic systems with general eigenvalues are discussed for the Cauchy problem with essentially periodic small initial data or nonperiodic initial data. A result of Klainerman and Majda is thus extended to the general case.

scholarly journals A remark on global solutions to random 3D vorticity equations for small initial data

2019 ◽  
Vol 24 (8) ◽  
pp. 4021-4030 ◽  
Author(s):  
Michael Röckner ◽  
◽  
Rongchan Zhu ◽  
Xiangchan Zhu ◽  
◽  
...  
Keyword(s):  
Initial Data ◽  
Global Solutions ◽  
Small Initial Data ◽  
Vorticity Equations
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