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August  2019, 39(8): 4359-4398. doi: 10.3934/dcds.2019177

Weak dispersion for the Dirac equation on asymptotically flat and warped product spaces

1. 

Dipartimento di Matematica, Università degli studi di Padova, Via Trieste, 63 35131 Padova PD, Italy

2. 

École Polytechnique, CNRS Université Paris-Saclay, 91123 Palaiseau Cedex, France

* Corresponding author: Federico Cacciafesta

Received  June 2018 Revised  February 2019 Published  May 2019

Fund Project: The second author is supported by ANR-18-CE40-0028 project ESSED.

In this paper we prove local smoothing estimates for the Dirac equation on some non-flat manifolds; in particular, we will consider asymptotically flat and warped products metrics. The strategy of the proofs relies on the multiplier method.

Citation: Federico Cacciafesta, Anne-Sophie De Suzzoni. Weak dispersion for the Dirac equation on asymptotically flat and warped product spaces. Discrete & Continuous Dynamical Systems - A, 2019, 39 (8) : 4359-4398. doi: 10.3934/dcds.2019177
References:
[1]

D. Baskin, A Strichartz Estimate for de Sitter Space, The AMSI-ANU Workshop on Spectral Theory and Harmonic Analysis, Proc. Centre Math. Appl. Austral. Nat. Univ., vol. 44, Austral. Nat. Univ., Canberra, 2010, 97–104.  Google Scholar

[2]

D. Baskin and Ja red Wunsch, Resolvent estimates and local decay of waves on conic manifolds, J. Differential Geom., 95 (2013), 183-214.   Google Scholar

[3]

M. D. BlairH. F. Smith and C. D. Sogge, Strichartz estimates for the wave equation on manifolds with boundary, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 1817-1829.  doi: 10.1016/j.anihpc.2008.12.004.  Google Scholar

[4]

N. BoussaidP. D'Ancona and L. Fanelli, Virial identity and weak dispersion for the magnetic dirac equation, Journal de Mathématiques Pures et Appliquées, 95 (2011), 137-150.  doi: 10.1016/j.matpur.2010.10.004.  Google Scholar

[5]

N. Burq, Global Strichartz estimates for nontrapping geometries: About an article by H. F. Smith and C. D. Sogge, Comm. Partial Differential Equations, 28 (2003), 1675-1683.  doi: 10.1081/PDE-120024528.  Google Scholar

[6]

F. Cacciafesta, Virial identity and dispersive estimates for the n-dimensional Dirac equation, J. Math. Sci. Univ. Tokyo, 18 (2011), 441-463.   Google Scholar

[7]

F. Cacciafesta, Smoothing estimates for variable coefficients Schroedinger equation with electromagnetic potential, J. Math. Anal. Appl., 402 (2013), 286-296.  doi: 10.1016/j.jmaa.2013.01.040.  Google Scholar

[8]

F. Cacciafesta and P. D'Ancona, Endpoint estimates and global existence for the nonlinear Dirac equation with a potential, J. Differential Equations, 254 (2013), 2233-2260.  doi: 10.1016/j.jde.2012.12.002.  Google Scholar

[9]

F. CacciafestaP. D'Ancona and R. Lucá, Helmholtz and dispersive equations with variable coefficients on external domains, SIAM J. Math. Anal., 48 (2016), 1798-1832.  doi: 10.1137/15M103769X.  Google Scholar

[10]

F. Cacciafesta and Er ic Séré, Local smoothing estimates for the Dirac Coulomb equation in 2 and 3 dimensions, J. Funct. Anal., 271 (2016), 2339-2358.  doi: 10.1016/j.jfa.2016.04.003.  Google Scholar

[11]

P. R. Chernoff, Essential self-adjointness of powers of generators of hyperbolic equations, J. Functional Analysis, 12 (1973), 401-414.  doi: 10.1016/0022-1236(73)90003-7.  Google Scholar

[12]

P. D'Ancona and L. Fanelli, Decay estimates for the wave and Dirac equations with a magnetic potential, Comm. Pure Appl. Math., 60 (2007), 357-392.  doi: 10.1002/cpa.20152.  Google Scholar

[13]

P. D'Ancona and L. Fanelli, Strichartz and smoothing estimates of dispersive equations with magnetic potentials, Comm. Partial Differential Equations, 33 (2008), 1082-1112.  doi: 10.1080/03605300701743749.  Google Scholar

[14]

L. Fanelli and L. Vega, Magnetic virial identities, weak dispersion and Strichartz inequalities, Math. Ann., 344 (2009), 249-278.  doi: 10.1007/s00208-008-0303-7.  Google Scholar

[15]

V. Fock, Geometrization of the Dirac thory of electrons, Zeit. f. Phys, 57 (1929), 261-277.   Google Scholar

[16]

A. HassellT. Tao and J. Wunsch, Sharp Strichartz estimates on nontrapping asymptotically conic manifolds, American Journal of Mathematics, 128 (2006), 963-1024.   Google Scholar

[17]

J. Metcalfe and D. Tataru, Global parametrices and dispersive estimates for variable coefficient wave equations, Math. Ann., 353 (2012), 1183-1237.  doi: 10.1007/s00208-011-0714-8.  Google Scholar

[18] L. E. Parker and D. J. Toms, Quantum Field Theory in Curved Spacetime, Cambridge university press, Cambridge, 2009.  doi: 10.1017/CBO9780511813924.  Google Scholar
[19]

M. M. G. Ricci and T. Levi-Civita, Berichtigungen zum Aufsatz, Math. Ann., 54 (1901), 608. doi: 10.1007/BF01450726.  Google Scholar

[20]

H. F. Smith and C. D. Sogge, On the critical semilinear wave equation outside convex obstacles, J. Amer. Math. Soc., 8 (1995), 879-916.  doi: 10.2307/2152832.  Google Scholar

[21]

D. Tataru, Strichartz estimates for second order hyperbolic operators with nonsmooth coefficients. Ⅲ, J. Amer. Math. Soc., 15 (2002), 419–442 (electronic). doi: 10.1090/S0894-0347-01-00375-7.  Google Scholar

[22]

A. Vasy and J. Wunsch, Morawetz estimates for the wave equation at low frequency, Math. Ann., 355 (2013), 1221-1254.  doi: 10.1007/s00208-012-0817-x.  Google Scholar

show all references

References:
[1]

D. Baskin, A Strichartz Estimate for de Sitter Space, The AMSI-ANU Workshop on Spectral Theory and Harmonic Analysis, Proc. Centre Math. Appl. Austral. Nat. Univ., vol. 44, Austral. Nat. Univ., Canberra, 2010, 97–104.  Google Scholar

[2]

D. Baskin and Ja red Wunsch, Resolvent estimates and local decay of waves on conic manifolds, J. Differential Geom., 95 (2013), 183-214.   Google Scholar

[3]

M. D. BlairH. F. Smith and C. D. Sogge, Strichartz estimates for the wave equation on manifolds with boundary, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 1817-1829.  doi: 10.1016/j.anihpc.2008.12.004.  Google Scholar

[4]

N. BoussaidP. D'Ancona and L. Fanelli, Virial identity and weak dispersion for the magnetic dirac equation, Journal de Mathématiques Pures et Appliquées, 95 (2011), 137-150.  doi: 10.1016/j.matpur.2010.10.004.  Google Scholar

[5]

N. Burq, Global Strichartz estimates for nontrapping geometries: About an article by H. F. Smith and C. D. Sogge, Comm. Partial Differential Equations, 28 (2003), 1675-1683.  doi: 10.1081/PDE-120024528.  Google Scholar

[6]

F. Cacciafesta, Virial identity and dispersive estimates for the n-dimensional Dirac equation, J. Math. Sci. Univ. Tokyo, 18 (2011), 441-463.   Google Scholar

[7]

F. Cacciafesta, Smoothing estimates for variable coefficients Schroedinger equation with electromagnetic potential, J. Math. Anal. Appl., 402 (2013), 286-296.  doi: 10.1016/j.jmaa.2013.01.040.  Google Scholar

[8]

F. Cacciafesta and P. D'Ancona, Endpoint estimates and global existence for the nonlinear Dirac equation with a potential, J. Differential Equations, 254 (2013), 2233-2260.  doi: 10.1016/j.jde.2012.12.002.  Google Scholar

[9]

F. CacciafestaP. D'Ancona and R. Lucá, Helmholtz and dispersive equations with variable coefficients on external domains, SIAM J. Math. Anal., 48 (2016), 1798-1832.  doi: 10.1137/15M103769X.  Google Scholar

[10]

F. Cacciafesta and Er ic Séré, Local smoothing estimates for the Dirac Coulomb equation in 2 and 3 dimensions, J. Funct. Anal., 271 (2016), 2339-2358.  doi: 10.1016/j.jfa.2016.04.003.  Google Scholar

[11]

P. R. Chernoff, Essential self-adjointness of powers of generators of hyperbolic equations, J. Functional Analysis, 12 (1973), 401-414.  doi: 10.1016/0022-1236(73)90003-7.  Google Scholar

[12]

P. D'Ancona and L. Fanelli, Decay estimates for the wave and Dirac equations with a magnetic potential, Comm. Pure Appl. Math., 60 (2007), 357-392.  doi: 10.1002/cpa.20152.  Google Scholar

[13]

P. D'Ancona and L. Fanelli, Strichartz and smoothing estimates of dispersive equations with magnetic potentials, Comm. Partial Differential Equations, 33 (2008), 1082-1112.  doi: 10.1080/03605300701743749.  Google Scholar

[14]

L. Fanelli and L. Vega, Magnetic virial identities, weak dispersion and Strichartz inequalities, Math. Ann., 344 (2009), 249-278.  doi: 10.1007/s00208-008-0303-7.  Google Scholar

[15]

V. Fock, Geometrization of the Dirac thory of electrons, Zeit. f. Phys, 57 (1929), 261-277.   Google Scholar

[16]

A. HassellT. Tao and J. Wunsch, Sharp Strichartz estimates on nontrapping asymptotically conic manifolds, American Journal of Mathematics, 128 (2006), 963-1024.   Google Scholar

[17]

J. Metcalfe and D. Tataru, Global parametrices and dispersive estimates for variable coefficient wave equations, Math. Ann., 353 (2012), 1183-1237.  doi: 10.1007/s00208-011-0714-8.  Google Scholar

[18] L. E. Parker and D. J. Toms, Quantum Field Theory in Curved Spacetime, Cambridge university press, Cambridge, 2009.  doi: 10.1017/CBO9780511813924.  Google Scholar
[19]

M. M. G. Ricci and T. Levi-Civita, Berichtigungen zum Aufsatz, Math. Ann., 54 (1901), 608. doi: 10.1007/BF01450726.  Google Scholar

[20]

H. F. Smith and C. D. Sogge, On the critical semilinear wave equation outside convex obstacles, J. Amer. Math. Soc., 8 (1995), 879-916.  doi: 10.2307/2152832.  Google Scholar

[21]

D. Tataru, Strichartz estimates for second order hyperbolic operators with nonsmooth coefficients. Ⅲ, J. Amer. Math. Soc., 15 (2002), 419–442 (electronic). doi: 10.1090/S0894-0347-01-00375-7.  Google Scholar

[22]

A. Vasy and J. Wunsch, Morawetz estimates for the wave equation at low frequency, Math. Ann., 355 (2013), 1221-1254.  doi: 10.1007/s00208-012-0817-x.  Google Scholar

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