Sharp resolvent and time-decay estimates for dispersive equations on asymptotically Euclidean backgrounds

The purpose of this article is twofold. First we give a very robust method for proving sharp time decay estimates for the most classical three models of dispersive Partial Differential Equations, the wave, Klein-Gordon and Schr{\"o}dinger equations, on curved geometries, showing under very general assumptions the exact same decay as for the Euclidean case. Then we also extend these decay properties to the case of boundary value problems.

[1]  Andrew Hassell,et al.  Resolvent at low energy III: The spectral measure , 2010, 1009.3084.

[2]  Nicolas Burq,et al.  Contrôle de l'équation des plaques en présence d'obstacles strictement convexes , 1992 .

[3]  Carleman Estimates and Absence of Embedded Eigenvalues , 2005, math-ph/0508052.

[4]  A. Vasy,et al.  Positive commutators at the bottom of the spectrum , 2009, 0909.4583.

[5]  Danna Zhou,et al.  d. , 1840, Microbial pathogenesis.

[6]  M. Murata.  Asymptotic expansions in time for solutions of Schrödinger-type equations , 1982 .

[7]  M. Zworski,et al.  Quantum decay rates in chaotic scattering , 2007, 0706.3242.

[8]  H. Isozaki,et al.  A Remark on the Micro-local Resolvent Estimates for Two Body Schrödinger Operators , 1985 .

[9]  Arne Jensen,et al.  Spectral properties of Schrödinger operators and time-decay of the wave functions , 1979 .

[10]  A. Soffer,et al.  Decay estimates for Schrödinger operators , 1991 .

[11]  C. Gérard.  A proof of the abstract limiting absorption principle by energy estimates , 2008 .

[12]  William R. Green,et al.  Dispersive estimates for higher dimensional Schrödinger operators with threshold eigenvalues. II. The even dimensional case , 2014, 1409.6328.

[13]  J. Nash.  Continuity of Solutions of Parabolic and Elliptic Equations , 1958 .

[14]  D. Robert.  Asymptotique de la phase de diffusion à haute énergie pour des perturbations du second ordre du laplacien , 1992 .

[15]  Decay for the wave and Schrödinger evolutions on manifolds with conical ends, Part II , 2009 .

[16]  L. Hörmander.  The analysis of linear partial differential operators , 1990 .

[17]  Local Energy Decay for Several Evolution Equations on Asymptotically Euclidean Manifolds , 2012 .

[18]  N. Tzvetkov,et al.  On global Strichartz estimates for non-trapping metrics , 2006, math/0611705.

[19]  J. Rauch.  Local decay of scattering solutions to Schrödinger's equation , 1978 .

[20]  R. Danchin,et al.  Fourier Analysis and Nonlinear Partial Differential Equations , 2011 .

[21]  A. Jensen.  Propagation estimates for Schrödinger-type operators , 1985 .

[22]  A. Jensen,et al.  Multiple commutator estimates and resolvent smoothness in quantum scattering theory , 1984 .

[23]  J. Moser.  A Harnack inequality for parabolic di2erential equations , 1964 .

[24]  G. Lebeau.  Contrôle de l'équation de Schrödinger , 1992 .

[25]  Dr. M. G. Worster.  Methods of Mathematical Physics , 1947, Nature.

[26]  William R. Green,et al.  Dispersive Estimates for Higher Dimensional Schrödinger Operators with Threshold Eigenvalues I: The Odd Dimensional Case , 2014, 1409.6323.

[27]  Xue Ping Wang.  Asymptotic expansion in time of the Schrödinger group on conical manifolds , 2006 .

[28]  B. Ducomet,et al.  Decay of local energy for solutions of the free Schrödinger equation in exterior domains , 2018, Kyoto Journal of Mathematics.

[29]  Richard B. Melrose,et al.  Singularities of boundary value problems. I , 1978 .

[30]  J. Bouclet,et al.  Low Frequency Estimates and Local Energy Decay for Asymptotically Euclidean Laplacians , 2010, 1003.6016.

[31]  D. Tataru.  Local decay of waves on asymptotically flat stationary space-times , 2009, 0910.5290.

[32]  Thierry Coulhon,et al.  Ultracontractivity and Nash Type Inequalities , 1996 .

[33]  D. Stroock,et al.  Upper bounds for symmetric Markov transition functions , 1986 .

[34]  P. Grisvard.  Singularities in Boundary Value Problems , 1992 .

[35]  Mouez Dimassi,et al.  Spectral asymptotics in the semi-classical limit , 1999 .

[36]  G. Vodev.  Local energy decay of solutions to the wave equation for nontrapping metrics , 2004 .

[37]  Y. Tsutsumi.  Local energy decay of solutions to the free Schrodinger equation in exterior domains , 1984 .

[38]  Nicolas Burq,et al.  Semi-classical estimates for the resolvent in nontrapping geometries , 2002 .