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A vector field method for non-trapping, radiating spacetimes

Jesús Oliver

Mathematics Department, California State University, East Bay, 25800 Carlos Bee Blvd, Hayward, CA 94542, USA

E-mail Address: jesus.oliver@csueastbay.edu

https://doi.org/10.1142/S021989161650020XCited by:4 (Source: Crossref)

Abstract

We study the global decay properties of solutions to the linear wave equation in 1+3 dimensions on time-dependent, weakly asymptotically flat spacetimes. Assuming non-trapping of null geodesics and a local energy decay estimate, we prove that sufficiently regular solutions to this equation have bounded conformal energy. As an application we also show a conformal energy estimate with vector fields applied to the solution as well as a global $L^{\infty}$ $L^{\infty}$ decay bound in terms of a weighted norm on initial data. For solutions to the wave equation in these dynamical backgrounds, our results reduce the problem of establishing the classical pointwise decay rate $t^{- 3 / 2}$ $t^{- 3 / 2}$ in the interior and $t^{- 1}$ $t^{- 1}$ along outgoing null cones to simply proving that local energy decay holds.

Communicated by P. G. LeFloch

Keywords:

References

1. S. Alinhac, On the Morawetz–Keel–Smith–Sogge inequality for the wave equation on a curved back-ground, Publ. Res. Inst. Math. Sci. 42(3) (2006) 705–720. Crossref, Web of Science, Google Scholar
2. S. Alinhac, Geometric Analysis of Hyperbolic Differential Equations: An Introduction, London Mathematical Society Lecture Note Series, Vol. 374 (Cambridge University Press, 2010). Crossref, Google Scholar
3. L. Andersson and P. Blue, Uniform energy bound and asymptotics for the Maxwell field on a slowly rotating Kerr black hole exterior, preprint (2013), arXiv:math-ap/13102664. Google Scholar
4. D. Baskin, A. Vasy and J. Wunsch, Asymptotics of radiation fields in asymptotically Minkowski space, preprint (2012), arXiv:math-ap/12125141. Google Scholar
5. L. Bieri, S. Miao and S. Shahshahani, Asymptotic properties of solutions of the Maxwell–Klein–Gordon equation with small data, preprint (2014), arXiv:math-ap/14082550. Google Scholar
6. L. Bieri and N. Zipser, Extensions of the Stability Theorem of the Minkowski space in General Relativity, AMS/IP Studies in Advanced Mathematics, Vol. 45 (American Mathematical Society, 2009). Crossref, Google Scholar
7. P. Blue and A. Soffer, The wave equation on the Schwarzschild metric. II: Local decay for the spin-2 Regge–Wheeler equation, J. Math. Phys. 46(1) (2005) 012502. Crossref, Web of Science, Google Scholar
8. P. Blue and J. Sterbenz, Uniform decay of local energy and the semi-linear wave equation on Schwarzschild space, Comm. Math. Phys. 268(2) (2005) 481–504. Crossref, Web of Science, Google Scholar
9. H. Bondi, M. G. J. van der Burg and A. W. K. Metzner, Gravitational waves in general relativity VII: Waves from axi-symmetric isolated systems, Proc. Roy. Soc. Ser. A 269 (1962) 21–52. Crossref, Google Scholar
10. J. F. Bony and D. Häfner, The semilinear wave equation on asymptotically Euclidean manifolds, Comm. Partial Differential Equations 35(1) (2010) 23–67. Crossref, Web of Science, Google Scholar
11. D. Christodoulou and S. Klainerman, The Global Nonlinear Stability of the Minkowski Space, Princeton Mathematical Series, Vol. 41 (Princeton University Press, 1993). Google Scholar
12. M. Dafermos and I. Rodnianski, The red-shift effect and radiation decay on black hole spacetimes, Comm. Pure Appl. Math. 62(7) (2009) 859–919. Crossref, Web of Science, Google Scholar
13. M. Dafermos and I. Rodnianski, A new physical-space approach to decay for the wave equation with applications to black hole spacetimes, in XVIth Int. Cong. on Mathematical Physics (World Scientific Publisher, 2010), pp. 421–432. Google Scholar
14. M. Dafermos, I. Rodnianski and Y. Shlapentokh-Rothman, Decay for solutions of the wave equation on Kerr exterior spacetimes iii: The full subextremal case $| a | < m$ $| a | < m$ , preprint (2014), arXiv:math-ap/14027034. Google Scholar
15. M. Keel, H. F. Smith and C. D. Sogge, Almost global existence for some semilinear wave equations, J. Anal. Math. 87 (2002) 265–279. Crossref, Web of Science, Google Scholar
16. S. Klainerman, Uniform decay estimates and the Lorentz invariance of the classical wave equation, Comm. Pure Appl. Math. 38(3) (1985) 321–332. Crossref, Web of Science, Google Scholar
17. S. Klainerman, A commuting vectorfields approach to Strichartz-type inequalities and applications to quasi-linear wave equations, Int. Math. Res. Not. 5 (2001) 221–274. Crossref, Web of Science, Google Scholar
18. S. Klainerman and F. Nicolò, The Evolution Problem in General Relativity, Progress in Mathematical Physics, Vol. 25 (Birkhäuser, 2003). Crossref, Google Scholar
19. S. Klainerman and T. C. Sideris, On almost global existence for nonrelativistic wave equations in $3$ $3$ D, Comm. Pure Appl. Math. 49(3) (1996) 307–321. Crossref, Web of Science, Google Scholar
20. H. Lindblad, J. Metcalfe, C. D. Sogge, M. Tohaneanu and C. Wang, The Strauss conjecture on Kerr black hole backgrounds, preprint (2013), arXiv:math-ap/13044145. Google Scholar
21. H. Lindblad and I. Rodnianski, The global stability of Minkowski spacetime in harmonic gauge, Ann. of Math. 171(3) (2010) 1401–1477. Crossref, Web of Science, Google Scholar
22. H. Lindblad and J. Sterbenz, Global stability for charged-scalar fields on Minkowski space, Int. Math. Res. Pap., Article ID: 52976 (2006). Google Scholar
23. J. Luk, Improved decay for solutions to the linear wave equation on a Schwarzschild black hole, Ann. Henri Poincaré 11(5) (2010) 805–880. Crossref, Web of Science, Google Scholar
24. J. Luk, The null condition and global existence for nonlinear wave equations on slowly rotating Kerr spacetimes, preprint (2010), arXiv:math-ap/10094109. Google Scholar
25. J. Luk, A vector field method approach to improved decay for solutions to the wave equation on a slowly rotating Kerr black hole, Anal. PDE 5(3) (2012) 553–625. Crossref, Web of Science, Google Scholar
26. J. Metcalfe and C. D. Sogge, Long-time existence of quasilinear wave equations exterior to star-shaped obstacles via energy methods, SIAM J. Math. Anal. 38(1) (2006) 188–209. Crossref, Web of Science, Google Scholar
27. J. Metcalfe and D. Tataru, Global parametrices and dispersive estimates for variable coefficient wave equations, Math. Ann. 353(4) (2012) 1183–1237. Crossref, Web of Science, Google Scholar
28. J. Metcalfe, D. Tataru and M. Tohaneanu, Price’s law on nonstationary spacetimes, Adv. Math. 230(3) (2012) 995–1028. Crossref, Web of Science, Google Scholar
29. C. S. Morawetz, The limiting amplitude principle, Comm. Pure Appl. Math. 15 (1962) 349–361. Crossref, Web of Science, Google Scholar
30. C. S. Morawetz, Time decay for the nonlinear Klein–Gordon equations, Proc. Roy. Soc. Ser. A 306 (1968) 291–296. Crossref, Google Scholar
31. J. Oliver and J. Sterbenz, A vector field method for black hole spacetimes, in preparation. Google Scholar
32. J. V. Ralston, Solutions of the wave equation with localized energy, Comm. Pure Appl. Math. 22 (1969) 807–823. Crossref, Web of Science, Google Scholar
33. I. Rodnianski and T. Tao, Effective limiting absorption principles, and applications, preprint (2011), arXiv:math-ap/1105.0873. Google Scholar
34. R. K. Sachs, Gravitational waves in general relativity VIII: Waves in asymptotically flat spacetime, Proc. Roy. Soc. Ser. A 270 (1962) 103–126. Crossref, Google Scholar
35. C. D. Sogge and C. Wang, Concerning the wave equation on asymptotically Euclidean manifolds, J. Anal. Math. 112 (2010) 1–32. Crossref, Web of Science, Google Scholar
36. J. Sterbenz, Angular regularity and Strichartz estimates for the wave equation, Int. Math. Res. Not. 4 (2005) 187–231, with an appendix by Igor Rodnianski. Crossref, Web of Science, Google Scholar
37. D. Tataru and M. Tohaneanu, A local energy estimate on Kerr black hole backgrounds, Int. Math. Res. Not. 2 (2011) 248–292. Google Scholar
38. S. Yang, Global solutions of nonlinear wave equations in time dependent inhomogeneous media, Arch. Ration. Mech. Anal. 209(2) (2013) 683–728. Crossref, Web of Science, Google Scholar
39. S. Yang, On the quasilinear wave equations in time dependent inhomogeneous media, preprint (2013), arXiv:math-ap/13127264. Google Scholar