American Institute of Mathematical Sciences

Advanced
July  2010, 28(3): 1007-1031. doi: 10.3934/dcds.2010.28.1007

On emerging scarred surfaces for the Einstein vacuum equations

Sergiu Klainerman 1, and  Igor Rodnianski 2,

1. 

Department of Mathematics, Princeton University, Princeton, NJ 08544,, United States
2. 

Department of Mathematics, Princeton University, Princeton, NJ 08544, United States

Received  March 2010 Revised  April 2010 Published  April 2010

We follow up our work [4] concerning the formation of trapped surfaces. We provide a considerable extension of our result there on pre-scared surfaces to allow for the formation of a surface with multiple pre-scared angular regions which, together, can cover an arbitrarily large portion of the surface. In a forthcoming paper we plan to show that once a significant part of the surface is pre-scared, it can be additionally deformed to produce a bona-fide trapped surface.
Keywords: Ricci coefficients, vacuum, characteristic., black hole, expansion, trapped surface, scarred surface, double null foliation, Einstein equations, null second fundamental form, energy estimates.
Mathematics Subject Classification: 35J1.
Citation: Sergiu Klainerman, Igor Rodnianski. On emerging scarred surfaces for the Einstein vacuum equations. Discrete and Continuous Dynamical Systems, 2010, 28 (3) : 1007-1031. doi: 10.3934/dcds.2010.28.1007
[1]

Genni Fragnelli. Null controllability of degenerate parabolic equations in non divergence form via Carleman estimates. Discrete and Continuous Dynamical Systems - S, 2013, 6 (3) : 687-701. doi: 10.3934/dcdss.2013.6.687
[2]

Fengyan Yang. Exact boundary null controllability for a coupled system of plate equations with variable coefficients. Evolution Equations and Control Theory, 2021  doi: 10.3934/eect.2021036
[3]

Walter Allegretto, Liqun Cao, Yanping Lin. Multiscale asymptotic expansion for second order parabolic equations with rapidly oscillating coefficients. Discrete and Continuous Dynamical Systems, 2008, 20 (3) : 543-576. doi: 10.3934/dcds.2008.20.543
[4]

El Mustapha Ait Ben Hassi, Farid Ammar khodja, Abdelkarim Hajjaj, Lahcen Maniar. Carleman Estimates and null controllability of coupled degenerate systems. Evolution Equations and Control Theory, 2013, 2 (3) : 441-459. doi: 10.3934/eect.2013.2.441
[5]

Farid Ammar Khodja, Cherif Bouzidi, Cédric Dupaix, Lahcen Maniar. Null controllability of retarded parabolic equations. Mathematical Control and Related Fields, 2014, 4 (1) : 1-15. doi: 10.3934/mcrf.2014.4.1
[6]

Lahcen Maniar, Martin Meyries, Roland Schnaubelt. Null controllability for parabolic equations with dynamic boundary conditions. Evolution Equations and Control Theory, 2017, 6 (3) : 381-407. doi: 10.3934/eect.2017020
[7]

Lydia Ouaili. Minimal time of null controllability of two parabolic equations. Mathematical Control and Related Fields, 2020, 10 (1) : 89-112. doi: 10.3934/mcrf.2019031
[8]

Phoebus Rosakis. Continuum surface energy from a lattice model. Networks and Heterogeneous Media, 2014, 9 (3) : 453-476. doi: 10.3934/nhm.2014.9.453
[9]

Marcelo M. Disconzi, Igor Kukavica. A priori estimates for the 3D compressible free-boundary Euler equations with surface tension in the case of a liquid. Evolution Equations and Control Theory, 2019, 8 (3) : 503-542. doi: 10.3934/eect.2019025
[10]

J. Carmelo Flores, Luz De Teresa. Null controllability of one dimensional degenerate parabolic equations with first order terms. Discrete and Continuous Dynamical Systems - B, 2020, 25 (10) : 3963-3981. doi: 10.3934/dcdsb.2020136
[11]

Piermarco Cannarsa, Patrick Martinez, Judith Vancostenoble. Persistent regional null contrillability for a class of degenerate parabolic equations. Communications on Pure and Applied Analysis, 2004, 3 (4) : 607-635. doi: 10.3934/cpaa.2004.3.607
[12]

Qi Lü, Enrique Zuazua. Robust null controllability for heat equations with unknown switching control mode. Discrete and Continuous Dynamical Systems, 2014, 34 (10) : 4183-4210. doi: 10.3934/dcds.2014.34.4183
[13]

Enrique Fernández-Cara, Luz de Teresa. Null controllability of a cascade system of parabolic-hyperbolic equations. Discrete and Continuous Dynamical Systems, 2004, 11 (2&3) : 699-714. doi: 10.3934/dcds.2004.11.699
[14]

Lin Yan, Bin Wu. Null controllability for a class of stochastic singular parabolic equations with the convection term. Discrete and Continuous Dynamical Systems - B, 2022, 27 (6) : 3213-3240. doi: 10.3934/dcdsb.2021182
[15]

Yuan Gao, Hangjie Ji, Jian-Guo Liu, Thomas P. Witelski. A vicinal surface model for epitaxial growth with logarithmic free energy. Discrete and Continuous Dynamical Systems - B, 2018, 23 (10) : 4433-4453. doi: 10.3934/dcdsb.2018170
[16]

Pedro J. Torres, R. Carretero-González, S. Middelkamp, P. Schmelcher, Dimitri J. Frantzeskakis, P.G. Kevrekidis. Vortex interaction dynamics in trapped Bose-Einstein condensates. Communications on Pure and Applied Analysis, 2011, 10 (6) : 1589-1615. doi: 10.3934/cpaa.2011.10.1589
[17]

Wenjun Wang, Lei Yao. Spherically symmetric Navier-Stokes equations with degenerate viscosity coefficients and vacuum. Communications on Pure and Applied Analysis, 2010, 9 (2) : 459-481. doi: 10.3934/cpaa.2010.9.459
[18]

Chengchun Hao. Cauchy problem for viscous shallow water equations with surface tension. Discrete and Continuous Dynamical Systems - B, 2010, 13 (3) : 593-608. doi: 10.3934/dcdsb.2010.13.593
[19]

Giovanni De Matteis, Gianni Manno. Lie algebra symmetry analysis of the Helfrich and Willmore surface shape equations. Communications on Pure and Applied Analysis, 2014, 13 (1) : 453-481. doi: 10.3934/cpaa.2014.13.453
[20]

Guido Cavallaro, Roberto Garra, Carlo Marchioro. Long time localization of modified surface quasi-geostrophic equations. Discrete and Continuous Dynamical Systems - B, 2021, 26 (9) : 5135-5148. doi: 10.3934/dcdsb.2020336

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (87)
  • HTML views (0)
  • Cited by (7)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy