Resolvent estimates with mild trapping

Wunsch, Jared

Journées équations aux dérivées partielles (2012), article no. 13, 15 p.

We discuss recent progress in understanding the effects of certain trapping geometries on cut-off resolvent estimates, and thus on the qualititative behavior of linear evolution equations. We focus on trapping that is unstable, so that strong resolvent estimates hold on the real axis, and large resonance-free regions can be shown to exist beyond it.

[1] Lars Andersson and Pieter Blue. Hidden symmetries and decay for the wave equation on the Kerr spacetime, 2009.

[2] Claude Bardos, Gilles Lebeau, and Jeffrey Rauch. Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary. SIAM J. Control Optim., 30(5):1024–1065, 1992. | MR | Zbl

[3] D. Baskin and J. Wunsch. Resolvent estimates and local decay of waves on conic manifolds. In preparation.

[4] Jean-François Bony, Nicolas Burq, and Thierry Ramond. Minoration de la résolvante dans le cas captif. C. R. Math. Acad. Sci. Paris, 348(23-24):1279–1282, 2010. | MR | Zbl

[5] N. Burq. Smoothing effect for Schrödinger boundary value problems. Duke Math. J., 123(2):403–427, 2004. | MR | Zbl

[6] N. Burq and H. Christianson. Imperfect geometric control and overdamping for the damped wave equation. In preparation.

[7] Nicolas Burq. Pôles de diffusion engendrés par un coin. Astérisque, (242):ii+122, 1997. | Numdam | MR | Zbl

[8] Nicolas Burq. Décroissance de l’énergie locale de l’équation des ondes pour le problème extérieur et absence de résonance au voisinage du réel. Acta Math., 180(1):1–29, 1998. | MR | Zbl

[9] H. Christianson, E. Schenck, A. Vasy, and J. Wunsch. From resolvent estimates to damped waves. Preprint, 2012.

[10] H. Christianson and J. Wunsch. Local smoothing for the Schrödinger equation with a prescribed loss. Amer. J. Math., to appear.

[11] Hans Christianson. Cutoff resolvent estimates and the semilinear Schrödinger equation. Proc. Amer. Math. Soc., 136(10):3513–3520, 2008. | MR | Zbl

[12] Hans Christianson. Dispersive estimates for manifolds with one trapped orbit. Comm. Partial Differential Equations, 33(7-9):1147–1174, 2008. | MR | Zbl

[13] Peter Constantin and Jean-Claude Saut. Effets régularisants locaux pour des équations dispersives générales. C. R. Acad. Sci. Paris Sér. I Math., 304(14):407–410, 1987. | MR | Zbl

[14] Mihalis Dafermos and Igor Rodnianski. Decay for solutions of the wave equation on kerr exterior spacetimes i-ii: The cases $\left|a\right|\ll m$ or axisymmetry, 2010.

[15] K. Datchev and A. Vasy. Semiclassical resolvent estimates at trapped sets. Preprint, 2012. | Numdam | MR | Zbl

[16] K. Datchev and A. Vasy. Gluing semiclassical resolvent estimates via propagation of singularities. IMRN, to appear, Preprint, 2011. | MR

[17] Kiril Datchev. Local smoothing for scattering manifolds with hyperbolic trapped sets. Comm. Math. Phys., 286(3):837–850, 2009. | MR | Zbl

[18] Shin-ichi Doi. Smoothing effects of Schrödinger evolution groups on Riemannian manifolds. Duke Math. J., 82(3):679–706, 1996. | MR | Zbl

[19] Roland Donninger, Wilhelm Schlag, and Avy Soffer. On pointwise decay of linear waves on a Schwarzschild black hole background. Comm. Math. Phys., 309(1):51–86, 2012. | MR | Zbl

[20] Semyon Dyatlov. Exponential energy decay for Kerr–de Sitter black holes beyond event horizons. Math. Res. Lett., 18(5):1023–1035, 2011. | MR | Zbl

[21] Semyon Dyatlov. Quasi-normal modes and exponential energy decay for the Kerr-de Sitter black hole. Comm. Math. Phys., 306(1):119–163, 2011. | MR | Zbl

[22] Neil Fenichel. Persistence and smoothness of invariant manifolds for flows. Indiana Univ. Math. J., 21:193–226, 1971/1972. | MR | Zbl

[23] 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(2):465–503, 2006. | MR | Zbl

[24] F. Finster, N. Kamran, J. Smoller, and S.-T. Yau. Erratum: “Decay of solutions of the wave equation in the Kerr geometry” [Comm. Math. Phys. 264 (2006), no. 2, 465–503;]. Comm. Math. Phys., 280(2):563–573, 2008. | MR | Zbl

[25] C. Gérard and J. Sjöstrand. Resonances en limite semiclassique et exposants de Lyapunov. Comm. Math. Phys., 116(2):193–213, 1988. | MR | Zbl

[26] M. W. Hirsch, C. C. Pugh, and M. Shub. Invariant manifolds. Springer-Verlag, Berlin, 1977. Lecture Notes in Mathematics, Vol. 583. | MR | Zbl

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

[28] Tosio Kato and Kenji Yajima. Some examples of smooth operators and the associated smoothing effect. Rev. Math. Phys., 1(4):481–496, 1989. | MR | Zbl

[29] P.D. Lax and R.S. Phillips. Scattering Theory. Academic Press, New York, 1967. Revised edition, 1989. | MR | Zbl

[30] G. Lebeau. Équation des ondes amorties. In Algebraic and geometric methods in mathematical physics (Kaciveli, 1993), pages 73–109. Kluwer Acad. Publ., Dordrecht, 1996. | MR | Zbl

[31] Richard Melrose and Jared Wunsch. Propagation of singularities for the wave equation on conic manifolds. Invent. Math., 156(2):235–299, 2004. | MR | Zbl

[32] Stéphane Nonnenmacher and Maciej Zworski. Quantum decay rates in chaotic scattering. Acta Math., 203(2):149–233, 2009. | MR | Zbl

[33] Stéphane Nonnenmacher and Maciej Zworski. Semiclassical resolvent estimates in chaotic scattering. Appl. Math. Res. Express. AMRX, (1):74–86, 2009. | MR | Zbl

[34] James V. Ralston. Solutions of the wave equation with localized energy. Comm. Pure Appl. Math., 22:807–823, 1969. | MR | Zbl

[35] Jeffrey Rauch and Michael Taylor. Decaying states of perturbed wave equations. J. Math. Anal. Appl., 54(1):279–285, 1976. | MR | Zbl

[36] T. Regge. Analytic properties of the scattering matrix. Nuovo Cimento (10), 8:671–679, 1958. | MR | Zbl

[37] Per Sjölin. Regularity of solutions to the Schrödinger equation. Duke Math. J., 55(3):699–715, 1987. | MR | Zbl

[38] J. Sjöstrand and M. Zworski. Complex scaling and the distribution of scattering poles. J. Amer. Math. Soc., 4:729–769, 1991. | MR | Zbl

[39] P. Stefanov and G. Vodev. Distribution of resonances for the Neumann problem in linear elasticity outside a strictly convex body. Duke Math. J., 78(3):677–714, 1995. | MR | Zbl

[40] P. Stefanov and G. Vodev. Neumann resonances in linear elasticity for an arbitrary body. Comm. Math. Phys., 176(3):645–659, 1996. | MR | Zbl

[41] Plamen Stefanov. Quasimodes and resonances: sharp lower bounds. Duke Math. J., 99(1):75–92, 1999. | MR | Zbl

[42] Siu-Hung Tang and Maciej Zworski. From quasimodes to resonances. Math. Res. Lett., 5(3):261–272, 1998. | MR | Zbl

[43] Siu-Hung Tang and Maciej Zworski. Resonance expansions of scattered waves. Comm. Pure Appl. Math., 53(10):1305–1334, 2000. | MR | Zbl

[44] Daniel Tataru. Local decay of waves on asymptotically flat stationary space-times. arXiv:0910.5290. | MR | Zbl

[45] Daniel Tataru and Mihai Tohaneanu. A local energy estimate on Kerr black hole backgrounds. Int. Math. Res. Not. IMRN, (2):248–292, 2011. | MR | Zbl

[46] B. R. Vaĭnberg. Exterior elliptic problems that depend polynomially on the spectral parameter, and the asymptotic behavior for large values of the time of the solutions of nonstationary problems. Mat. Sb. (N.S.), 92(134):224–241, 343, 1973. | MR | Zbl

[47] B. R. Vaĭnberg. Asymptotic methods in equations of mathematical physics. Gordon & Breach Science Publishers, New York, 1989. Translated from the Russian by E. Primrose. | MR | Zbl

[48] A. Vasy. Microlocal analysis of asymptotically hyperbolic and kerr-de sitter spaces. Preprint, 2010.

[49] Luis Vega. Schrödinger equations: pointwise convergence to the initial data. Proc. Amer. Math. Soc., 102(4):874–878, 1988. | MR | Zbl

[50] Jared Wunsch and Maciej Zworski. Resolvent estimates for normally hyperbolic trapped sets. Ann. Henri Poincaré, 12(7):1349–1385, 2011. | MR | Zbl

[51] Kenji Yajima. On smoothing property of Schrödinger propagators. In Functional-analytic methods for partial differential equations (Tokyo, 1989), volume 1450 of Lecture Notes in Math., pages 20–35. Springer, Berlin, 1990. | MR | Zbl

[52] M. Zworski. Personal communication.

[53] M. Zworski. Distribution of poles for scattering on the real line. J. Funct. Anal., 73:277–296, 1987. | MR | Zbl

[54] Maciej Zworski. Resonances in physics and geometry. Notices Amer. Math. Soc., 46(3):319–328, 1999. | MR | Zbl