Quasilinear waves and trapping: Kerr-de Sitter space
Hintz, Peter1 ; Vasy, András1
1 Department of Mathematics Stanford University CA 94305-2125, USA
Journées équations aux dérivées partielles (2014), article no. 10, 15 p.

In these notes, we will describe recent work on globally solving quasilinear wave equations in the presence of trapped rays, on Kerr-de Sitter space, and obtaining the asymptotic behavior of solutions. For the associated linear problem without trapping, one would consider a global, non-elliptic, Fredholm framework; in the presence of trapping the same framework is available for spaces of growing functions only. In order to solve the quasilinear problem we thus combine these frameworks with the normally hyperbolic trapping results of Dyatlov and a Nash-Moser iteration scheme.

DOI : 10.5802/jedp.113
Hintz, Peter 1 ; Vasy, András 1

1 Department of Mathematics Stanford University CA 94305-2125, USA 
@article{JEDP_2014____A10_0,
     author = {Hintz, Peter and Vasy, Andr\'as},
     title = {Quasilinear waves and trapping: {Kerr-de} {Sitter} space},
     journal = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
     eid = {10},
     pages = {1--15},
     publisher = {Groupement de recherche 2434 du CNRS},
     year = {2014},
     doi = {10.5802/jedp.113},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/jedp.113/}
}
TY  - JOUR
AU  - Hintz, Peter
AU  - Vasy, András
TI  - Quasilinear waves and trapping: Kerr-de Sitter space
JO  - Journées équations aux dérivées partielles
PY  - 2014
SP  - 1
EP  - 15
PB  - Groupement de recherche 2434 du CNRS
UR  - http://www.numdam.org/articles/10.5802/jedp.113/
DO  - 10.5802/jedp.113
LA  - en
ID  - JEDP_2014____A10_0
ER  - 
%0 Journal Article
%A Hintz, Peter
%A Vasy, András
%T Quasilinear waves and trapping: Kerr-de Sitter space
%J Journées équations aux dérivées partielles
%D 2014
%P 1-15
%I Groupement de recherche 2434 du CNRS
%U http://www.numdam.org/articles/10.5802/jedp.113/
%R 10.5802/jedp.113
%G en
%F JEDP_2014____A10_0
Hintz, Peter; Vasy, András. Quasilinear waves and trapping: Kerr-de Sitter space. Journées équations aux dérivées partielles (2014), article  no. 10, 15 p. doi : 10.5802/jedp.113. http://www.numdam.org/articles/10.5802/jedp.113/

[1] Andersson, L.; Blue, P. Uniform energy bound and asymptotics for the Maxwell field on a slowly rotating Kerr black hole exterior, Preprint, arXiv:1310.2664 (2013)

[2] Bachelot, A. Gravitational scattering of electromagnetic field by Schwarzschild black-hole, Ann. Inst. H. Poincaré Phys. Théor., Volume 54 (1991) no. 3, pp. 261-320 | Numdam | MR | Zbl

[3] Bachelot, A. Scattering of electromagnetic field by de Sitter-Schwarzschild black hole, Nonlinear hyperbolic equations and field theory (Lake Como, 1990) (Pitman Res. Notes Math. Ser.), Volume 253, Longman Sci. Tech., Harlow, 1992, pp. 23-35 | MR | Zbl

[4] Barreto, A. Sá; Zworski, M. Distribution of resonances for spherical black holes, Math. Res. Lett., Volume 4 (1997) no. 1, pp. 103-121 | MR | Zbl

[5] Beals, M.; Reed, M. Microlocal regularity theorems for nonsmooth pseudodifferential operators and applications to nonlinear problems, Trans. Amer. Math. Soc., Volume 285 (1984) no. 1, pp. 159-184 | DOI | MR | Zbl

[6] Blue, P.; Soffer, A. Phase space analysis on some black hole manifolds, J. Funct. Anal., Volume 256 (2009) no. 1, pp. 1-90 | DOI | MR | Zbl

[7] Bony, J.-F.; Häfner, D. Decay and non-decay of the local energy for the wave equation on the de Sitter-Schwarzschild metric, Comm. Math. Phys., Volume 282 (2008) no. 3, pp. 697-719 | MR | Zbl

[8] Carter, B. Global structure of the Kerr family of gravitational fields, Phys. Rev., Volume 174 (1968), pp. 1559-1571 | Zbl

[9] Carter, B. Hamilton-Jacobi and Schrödinger separable solutions of Einstein’s equations, Comm. Math. Phys., Volume 10 (1968), pp. 280-310 | MR | Zbl

[10] Dafermos, M.; Holzegel, G.; Rodnianski, I. A scattering theory construction of dynamical vacuum black holes, Preprint, arxiv:1306.5364 (2013)

[11] Dafermos, M.; Rodnianski, I. A proof of Price’s law for the collapse of a self-gravitating scalar field, Invent. Math., Volume 162 (2005) no. 2, pp. 381-457 | MR | Zbl

[12] Dafermos, M.; Rodnianski, I. The wave equation on Schwarzschild-de Sitter space times, Preprint, arXiv:07092766 (2007)

[13] Dafermos, M.; Rodnianski, I. The red-shift effect and radiation decay on black hole spacetimes, Comm. Pure Appl. Math, Volume 62 (2009), pp. 859-919 | MR | Zbl

[14] Dafermos, M.; Rodnianski, I. Decay of solutions of the wave equation on Kerr exterior space-times I-II: The cases of |a|≪M or axisymmetry, Preprint, arXiv:1010.5132 (2010) | MR

[15] Dafermos, M.; Rodnianski, I.; et al, T. Damour The black hole stability problem for linear scalar perturbations, Proceedings of the Twelfth Marcel Grossmann Meeting on General Relativity, World Scientific, Singapore (2011), pp. 132-189 (arXiv:1010.5137)

[16] Dafermos, M.; Rodnianski, I. Lectures on black holes and linear waves, Evolution equations (Clay Math. Proc.), Volume 17, Amer. Math. Soc., Providence, RI, 2013, pp. 97-205 | MR

[17] Dafermos, M.; Rodnianski, I.; Shlapentokh-Rothman, Y. Decay for solutions of the wave equation on Kerr exterior spacetimes III: The full subextremal case |a|<M, Preprint, arXiv:1402.7034 (2014)

[18] Donninger, R.; Schlag, W.; Soffer, A. A proof of Price’s law on Schwarzschild black hole manifolds for all angular momenta, Adv. Math., Volume 226 (2011) no. 1, pp. 484-540 | DOI | MR | Zbl

[19] Dyatlov, S. Exponential energy decay for Kerr–de Sitter black holes beyond event horizons, Math. Res. Lett., Volume 18 (2011) no. 5, pp. 1023-1035 | MR | Zbl

[20] Dyatlov, S. Quasi-normal modes and exponential energy decay for the Kerr-de Sitter black hole, Comm. Math. Phys., Volume 306 (2011) no. 1, pp. 119-163 | DOI | MR | Zbl

[21] Dyatlov, S. Asymptotics of linear waves and resonances with applications to black holes, Preprint, arXiv:1305.1723 (2013)

[22] Dyatlov, S. Resonance projectors and asymptotics for r-normally hyperbolic trapped sets, Preprint, arXiv:1301.5633 (2013)

[23] Dyatlov, S. Spectral gaps for normally hyperbolic trapping, Preprint, arXiv:1403.6401 (2013)

[24] Dyatlov, S.; Zworski, M. Trapping of waves and null geodesics for rotating black holes, Phys. Rev. D, Volume 88 (2013), pp. 084037

[25] Finster, F.; Kamran, N.; Smoller, J.; Yau, S.-T. Decay of solutions of the wave equation in the Kerr geometry, Comm. Math. Phys., Volume 264 (2006) no. 2, pp. 465-503 | DOI | MR | Zbl

[26] Finster, F.; Kamran, N.; Smoller, J.; Yau, S.-T. Linear waves in the Kerr geometry: a mathematical voyage to black hole physics, Bull. Amer. Math. Soc. (N.S.), Volume 46 (2009) no. 4, pp. 635-659 | DOI | MR | Zbl

[27] Hintz, P. Global well-posedness of quasilinear wave equations on asymptotically de Sitter spaces, Preprint, arXiv:1311.6859 (2013)

[28] Hintz, P.; Vasy, A. Non-trapping estimates near normally hyperbolic trapping, Preprint, arXiv:1306.4705 (2013)

[29] Hintz, P.; Vasy, A. Semilinear wave equations on asymptotically de Sitter, Kerr-de Sitter and Minkowski spacetimes, Preprint, arXiv:1306.4705 (2013)

[30] Hintz, P.; Vasy, A. Global analysis of quasilinear wave equations on asymptotically Kerr-de Sitter spaces, Preprint, arXiv:1404.1348 (2014)

[31] Hörmander, L. On the existence and the regularity of solutions of linear pseudo-differential equations, Enseignement Math. (2), Volume 17 (1971), pp. 99-163 | MR | Zbl

[32] Hörmander, L. The analysis of linear partial differential operators, vol. 1-4, Springer-Verlag, 1983 | Zbl

[33] Kay, B. S.; Wald, R. M. Linear stability of Schwarzschild under perturbations which are nonvanishing on the bifurcation 2-sphere, Classical Quantum Gravity, Volume 4 (1987) no. 4, pp. 893-898 http://stacks.iop.org/0264-9381/4/893 | MR | Zbl

[34] Luk, J. The null condition and global existence for nonlinear wave equations on slowly rotating Kerr spacetimes, J. Eur. Math. Soc. (JEMS), Volume 15 (2013) no. 5, pp. 1629-1700 | DOI | MR | Zbl

[35] Marzuola, J.; Metcalfe, J.; Tataru, D.; Tohaneanu, M. Strichartz estimates on Schwarzschild black hole backgrounds, Comm. Math. Phys., Volume 293 (2010) no. 1, pp. 37-83 | DOI | MR | Zbl

[36] Melrose, R. B. Transformation of boundary problems, Acta Math., Volume 147 (1981) no. 3-4, pp. 149-236 | MR | Zbl

[37] Melrose, R. B. The Atiyah-Patodi-Singer index theorem, Research Notes in Mathematics, 4, A K Peters Ltd., Wellesley, MA, 1993, pp. xiv+377 | MR | Zbl

[38] Melrose, R. B.; Ikawa, M. Spectral and scattering theory for the Laplacian on asymptotically Euclidian spaces, Marcel Dekker (1994) | MR | Zbl

[39] Melrose, R. B.; Barreto, A. Sá; Vasy, A. Asymptotics of solutions of the wave equation on de Sitter-Schwarzschild space, Comm. in PDEs, Volume 39 (2014) no. 3, pp. 512-529 | MR | Zbl

[40] Nonnenmacher, S.; Zworski, M. Decay of correlations for normally hyperbolic trapping, Preprint, arXiv:1302.4483 (2013)

[41] Raymond, X. Saint A simple Nash-Moser implicit function theorem, Enseign. Math. (2), Volume 35 (1989) no. 3-4, pp. 217-226 | MR | Zbl

[42] Tataru, D. Local decay of waves on asymptotically flat stationary space-times, Amer. J. Math., Volume 135 (2013) no. 2, pp. 361-401 | DOI | MR | Zbl

[43] Tataru, D.; Tohaneanu, M. A local energy estimate on Kerr black hole backgrounds, Int. Math. Res. Not. IMRN (2011) no. 2, pp. 248-292 | DOI | MR | Zbl

[44] Tohaneanu, M. Strichartz estimates on Kerr black hole backgrounds, Trans. Amer. Math. Soc., Volume 364 (2012) no. 2, pp. 689-702 | DOI | MR | Zbl

[45] Vasy, A. Microlocal analysis of asymptotically hyperbolic spaces and high energy resolvent estimates, MSRI Publications, 60, Cambridge University Press (2012) | MR

[46] Vasy, A. Microlocal analysis of asymptotically hyperbolic and Kerr-de Sitter spaces, Inventiones Math., Volume 194 (2013), pp. 381-513 (With an appendix by S. Dyatlov) | MR

[47] Wald, R. M. Note on the stability of the Schwarzschild metric, J. Math. Phys., Volume 20 (1979) no. 6, pp. 1056-1058 | DOI | MR

[48] Wunsch, J.; Zworski, M. Resolvent estimates for normally hyperbolic trapped sets, Ann. Henri Poincaré, Volume 12 (2011) no. 7, pp. 1349-1385 | DOI | MR | Zbl

[49] Yoshida, S.; Uchikata, N.; Futamase, T. Quasinormal modes of Kerr-de Sitter black holes, Phys. Rev. D, Volume 81 (2010) no. 4, pp. 044005, 14 | DOI | MR

Cité par Sources :