Multi-Center Vector Field Methods for Wave Equations
Tóm tắt
We develop the method of vector-fields to further study Dispersive Wave Equations. Radial vector fields are used to get a-priori estimates such as the Morawetz estimate on solutions of Dispersive Wave Equations. A key to such estimates is the repulsiveness or nontrapping conditions on the flow corresponding to the wave equation. Thus this method is limited to potential perturbations which are repulsive, that is the radial derivative pointing away from the origin. In this work, we generalize this method to include potentials which are repulsive relative to a line in space (in three or higher dimensions), among other cases. This method is based on constructing multi-centered vector fields as multipliers, cancellation lemmas and energy localization.
Tài liệu tham khảo
Amrein, WO, Boutet de Monvel, A, Georgescu, V, et al.: Commutator methods and spectral theory of N-body hamiltonian. Progress Math., 135 (1996)
Beceanu, M, Goldberg, M: Schrödinger dispersive estimates for a scaling-critical class of potentials. Commun. Math. Phys. 314(2), 471–481 (2012)
Blue, P, Soffer, A: Semilinear wave equations on the Schwarzschild manifold I: Local decay estimates. Adv. Diff. Equa. 8(gr-qc/0310091), 595–614 (2003)
Blue, P, Soffer, A: The wave equation on the schwarzschild metric ii. local decay for the spin-2 regge–wheeler equation. J. Math. Phys. 46(1), 012502 (2005)
Blue, P, Sterbenz, J: Uniform decay of local energy and the semi-linear wave equation on Schwarzschild space. Commun. Math. Phys. 268(2), 481–504 (2006)
Blue, P., Soffer, A.: A space–time integral estimate for a large data semi-linear wave equation on the Schwarzschild manifold. Lett. Math. Phys. 81(3), 227–238 (2007)
Blue, P., Soffer, A.: Phase space analysis on some black hole manifolds. J. Funct. Anal. 256(1), 1–90 (2009)
Dafermos, M., Rodnianski, I.: A proof of Price’s law for the collapse of a self-gravitating scalar field. Inventiones Mathematicae 162(2), 381–457 (2005)
Dafermos, M., Rodnianski, I.: A new physical-space approach to decay for the wave equation with applications to black hole spacetimes, pp 421–432 (2010)
Donninger, R, Schlag, W, Soffer, A: On pointwise decay of linear waves on a Schwarzschild black hole background. Commun. Math. Phys. 309(1), 51–86 (2012)
Graf, G. M.: Asymptotic completeness for N-body short-range quantum systems: A new proof. Commun. Math. Phys. 132(1), 73–101 (1990)
Hunziker, W., Sigal, I.M.: The quantum N-body problem. J. Math. Phys. 41(6), 3448 (2000)
Killip, R, Miao, C, Visan, M, Zhang, J, Zheng, J: The energy-critical NLS with inverse-square potential. arXiv:1509.05822 (2015)
Keel, M, Tao, T: Endpoint strichartz estimates. Amer. J. Math., 955–980 (1998)
Lavine, R. B: Commutators and scattering theory. Commun. Math. Phys. 20 (4), 301–323 (1971)
Lin, J.-E., Strauss, W.A: Decay and scattering of solutions of a nonlinear Schrödinger equation. J. Funct. Anal. 30(2), 245–263 (1978)
Luk, J.: 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), 553–625 (2012)
Morawetz, C.S: Time decay for the nonlinear Klein-Gordon equation. In: Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, vol. 306, pp. 291–296. The Royal Society (1968)
Morawetz, C.S., Ralston, J.V., Strauss, W.A.: Decay of solutions of the wave equation outside nontrapping obstacles. Commun. Pure Appl. Math. 30(4), 447–508 (1977)
Metcalfe, J., Tataru, D.: Decay estimates for variable coefficient wave equations in exterior domains. Adv. Phase Space Anal. Partial Diff. Equ., 201–216 (2009)
Soffer, A.: Monotonic local decay estimates. arXiv:1110.6549 (2011)
Sigal, I. M, Soffer, A.: The N-particle scattering problem: Asymptotic completeness for short-range systems. Ann. Math., 35–108 (1987)
Sigal, I M, Soffer, A: Local decay and propagation estimates for time dependent and time independent Hamiltonians. Preprint Princeton University (1988)
Tao, T.: Nonlinear dispersive equations: Local and global analysis (2006)
Tataru, D.: Parametrices and dispersive estimates for Schrödinger operators with variable coefficients. Amer. J. Math. 130(3), 571–634 (2008)