Strichartz estimates on Kerr black hole backgrounds
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- by Mihai Tohaneanu PDF
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Abstract:
We study the dispersive properties for the wave equation in the Kerr space-time with small angular momentum. The main result of this paper is to establish Strichartz estimates for solutions of the aforementioned equation. This follows a local energy decay result for the Kerr space-time obtained in earlier work of Tataru and the author, and uses the techniques and results by the author and collaborators (2010). As an application, we then prove global well-posedness and uniqueness for the energy critical semilinear wave equation with small initial data.References
- L. Andersson and P. Blue: Hidden symmetries and decay for the wave equation on the Kerr spacetime, arXiv:0908.2265v1
- Philip Brenner, On $L_{p}-L_{p^{\prime } }$ estimates for the wave-equation, Math. Z. 145 (1975), no. 3, 251–254. MR 387819, DOI 10.1007/BF01215290
- S. Chandrasekhar, The mathematical theory of black holes, International Series of Monographs on Physics, vol. 69, The Clarendon Press, Oxford University Press, New York, 1992. Revised reprint of the 1983 original; Oxford Science Publications. MR 1210321
- M. Dafermos and I. Rodnianski: A proof of the uniform boundedness of solutions to the wave equation on slowly rotating Kerr backgrounds, arXiv:0805.4309v1
- M. Dafermos and I. Rodnianski: Lectures on black holes and linear waves, arXiv:0811.0354v1
- F. Finster, N. Kamran, J. Smoller, and S.-T. Yau, Decay of solutions of the wave equation in the Kerr geometry, Comm. Math. Phys. 264 (2006), no. 2, 465–503. MR 2215614, DOI 10.1007/s00220-006-1525-8
- Felix Finster and Joel Smoller, A time-independent energy estimate for outgoing scalar waves in the Kerr geometry, J. Hyperbolic Differ. Equ. 5 (2008), no. 1, 221–255. MR 2405857, DOI 10.1142/S0219891608001453
- J. Ginibre and G. Velo, Generalized Strichartz inequalities for the wave equation, J. Funct. Anal. 133 (1995), no. 1, 50–68. MR 1351643, DOI 10.1006/jfan.1995.1119
- S. W. Hawking and G. F. R. Ellis, The large scale structure of space-time, Cambridge Monographs on Mathematical Physics, No. 1, Cambridge University Press, London-New York, 1973. MR 0424186
- L. V. Kapitanskiĭ, Some generalizations of the Strichartz-Brenner inequality, Algebra i Analiz 1 (1989), no. 3, 127–159 (Russian); English transl., Leningrad Math. J. 1 (1990), no. 3, 693–726. MR 1015129
- Markus Keel and Terence Tao, Endpoint Strichartz estimates, Amer. J. Math. 120 (1998), no. 5, 955–980. MR 1646048
- Hans Lindblad and Christopher D. Sogge, On existence and scattering with minimal regularity for semilinear wave equations, J. Funct. Anal. 130 (1995), no. 2, 357–426. MR 1335386, DOI 10.1006/jfan.1995.1075
- Jeremy Marzuola, Jason Metcalfe, Daniel Tataru, and Mihai Tohaneanu, Strichartz estimates on Schwarzschild black hole backgrounds, Comm. Math. Phys. 293 (2010), no. 1, 37–83. MR 2563798, DOI 10.1007/s00220-009-0940-z
- J. Metcalfe and D. Tataru: Global parametrices and dispersive estimates for variable coefficient wave equations, arXiv:0707.1191.
- Gerd Mockenhaupt, Andreas Seeger, and Christopher D. Sogge, Local smoothing of Fourier integral operators and Carleson-Sjölin estimates, J. Amer. Math. Soc. 6 (1993), no. 1, 65–130. MR 1168960, DOI 10.1090/S0894-0347-1993-1168960-6
- Hartmut Pecher, Nonlinear small data scattering for the wave and Klein-Gordon equation, Math. Z. 185 (1984), no. 2, 261–270. MR 731347, DOI 10.1007/BF01181697
- Hart F. Smith, A parametrix construction for wave equations with $C^{1,1}$ coefficients, Ann. Inst. Fourier (Grenoble) 48 (1998), no. 3, 797–835 (English, with English and French summaries). MR 1644105
- Robert S. Strichartz, A priori estimates for the wave equation and some applications, J. Functional Analysis 5 (1970), 218–235. MR 0257581, DOI 10.1016/0022-1236(70)90027-3
- Robert S. Strichartz, Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke Math. J. 44 (1977), no. 3, 705–714. MR 512086
- Daniel Tataru, Strichartz estimates for operators with nonsmooth coefficients and the nonlinear wave equation, Amer. J. Math. 122 (2000), no. 2, 349–376. MR 1749052
- Daniel Tataru, Strichartz estimates for second order hyperbolic operators with nonsmooth coefficients. II, Amer. J. Math. 123 (2001), no. 3, 385–423. MR 1833146
- Daniel Tataru, Strichartz estimates for second order hyperbolic operators with nonsmooth coefficients. III, J. Amer. Math. Soc. 15 (2002), no. 2, 419–442. MR 1887639, DOI 10.1090/S0894-0347-01-00375-7
- D. Tataru and M. Tohaneanu: Local energy estimate on Kerr black hole backgrounds, arXiv:0810.5766v2
Additional Information
- Mihai Tohaneanu
- Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907-2067
- Address at time of publication: Department of Mathematics, Johns Hopkins University, Baltimore, Maryland 21218
- Received by editor(s): January 8, 2010
- Published electronically: September 29, 2011
- © Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 364 (2012), 689-702
- MSC (2010): Primary 35Q75
- DOI: https://doi.org/10.1090/S0002-9947-2011-05405-X
- MathSciNet review: 2846348