Strichartz estimates on Kerr black hole backgrounds

Tohaneanu, Mihai

doi:10.1090/S0002-9947-2011-05405-X

Strichartz estimates on Kerr black hole backgrounds
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Trans. Amer. Math. Soc. 364 (2012), 689-702 Request permission

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