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2014, 13(6): 2177-2210. doi: 10.3934/cpaa.2014.13.2177

Strichartz estimates for Schrödinger equations with variable coefficients and unbounded potentials II. Superquadratic potentials

1. 

Department of Mathematics, Gakushuin University, 1-5-1 Mejiro, Toshima-ku, Tokyo 171-8588, Japan

Received  February 2013 Revised  January 2014 Published  July 2014

In this paper we prove local-in-time Strichartz estimates with loss of derivatives for Schrödinger equations with variable coefficients and potentials, under the conditions that the geodesic flow is nontrapping and potentials grow polynomially at infinity. This is a generalization to the case with variable coefficients and improvement of the result by Yajima-Zhang [40]. The proof is based on microlocal techniques including the semiclassical parametrix for a time scale depending on a spatial localization and the Littlewood-Paley type decomposition with respect to both of space and frequency.
Citation: Haruya Mizutani. Strichartz estimates for Schrödinger equations with variable coefficients and unbounded potentials II. Superquadratic potentials. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2177-2210. doi: 10.3934/cpaa.2014.13.2177
References:
[1]

J. -M. Bouclet, Strichartz estimates on asymptotically hyperbolic manifolds,, \emph{Anal. PDE.}, 4 (2011), 1. doi: 10.2140/apde.2011.4.1.

[2]

J. -M. Bouclet and N. Tzvetkov, Strichartz estimates for long range perturbations,, \emph{Amer. J. Math.}, 129 (2007), 1565. doi: 10.1353/ajm.2007.0039.

[3]

J. -M. Bouclet and N. Tzvetkov, On global Strichartz estimates for non trapping metrics,, \emph{J. Funct. Analysis}, 254 (2008), 1661. doi: 10.1016/j.jfa.2007.11.018.

[4]

N. Burq, P. Gérard and N. Tzvetkov, Strichartz inequalities and the nonlinear Schrödinger equation on compact manifolds,, \emph{Amer. J. Math.}, 126 (2004), 569.

[5]

N. Burq, C. Guillarmou and A. Hassell, Strichartz estimates without loss on manifolds with hyperbolic trapped geodesics,, \emph{Geom. Funct. Anal.}, 20 (2010), 627. doi: 10.1007/s00039-010-0076-5.

[6]

N. Burq, F. Planchon, J. Stalker and A. S. Tahvildar-Zadeh, Strichartz estimates for the wave and Schrödinger equations with potentials of critical decay,, \emph{Indiana Univ. Math. J.}, 53 (2004), 1665. doi: 10.1512/iumj.2004.53.2541.

[7]

T. Cazenave, Semilinear Schrödinger Equations,, Courant. Lect. Nates Math. vol. 10, (2003).

[8]

P. D'Ancona and L. Fanelli, Smoothing estimates for the Schrödinger equation with unbounded potentials,, \emph{J. Differential Equations}, 246 (2009), 4552. doi: 10.1016/j.jde.2009.03.026.

[9]

P. D'Ancona, L. Fanelli, L. Vega and N. Visciglia, Endpoint Strichartz estimates for the magnetic Schrödinger equation,, \emph{J. Funct. Analysis}, 258 (2010), 3227. doi: 10.1016/j.jfa.2010.02.007.

[10]

S. Doi, Smoothness of solutions for Schrödinger equations with unbounded potentials,, \emph{Publ. Res. Inst. Math. Sci.}, 41 (2005), 175.

[11]

D. Dunford and J. T. Schwartz, Linear operators. Part II. Spectral theory. Selfadjoint operators in Hilbert space. With the assistance of William G. Bade and Robert G. Bartle,, Interscience Publishers John Wiley & Sons New York-London, (1963).

[12]

L. Fanelli, V. Felli, M. Fontelos and M. A. Primo, Time decay of scaling critical electromagnetic Schrödinger flows,, \emph{Commun. Math. Phs.}, 324 (2013), 1033. doi: 10.1007/s00220-013-1830-y.

[13]

D. Fujiwara, Remarks on convergence of the Feynman path integrals,, \emph{Duke Math. J.}, 47 (1980), 559.

[14]

J. Ginibre and G. Velo, The global Cauchy problem for the non linear Schrödinger equation revisited,, \emph{Ann. lHP-Analyse non lin\'eaire.}, 2 (1985), 309.

[15]

A. Hassell, T. Tao and J. Wunsch, Sharp Strichartz estimates on nontrapping asymptotically conic manifolds,, \emph{Amer. J. Math.}, 128 (2006), 963.

[16]

B. Helffer and J. Sjöstrand, Equation de Schrödinger avec champ magnétique et équation de Harper,, in \emph{Schr\, 345 (1989), 118. doi: 10.1007/3-540-51783-9_19.

[17]

L. Hörmander, The Analysis of Linear Partial Differential Operators III,, Springer- Verlag, (1985).

[18]

A. Iwatsuka, Essential self-adjointness of the Schrödinger operators with magnetic fields diverging at infinity,, \emph{Publ. Res. Inst. Math. Sci.}, 26 (1990), 841. doi: 10.2977/prims/1195170737.

[19]

J.- L. Journé, A. Soffer, C. D. Sogge, Decay estimates for Schrödinger operators,, \emph{Comm. Pure Appl. Math.}, (1991), 573. doi: 10.1002/cpa.3160440504.

[20]

M. Keel and T. Tao, Endpoint Strichartz Estimates,, \emph{Amer. J. Math.}, 120 (1998), 955.

[21]

A. Martinez, An Introduction to Semiclassical and Microlocal Analysis,, Springer-Verlag, (2002). doi: 10.1007/978-1-4757-4495-8.

[22]

J. Marzuola, J. Metcalfe and D. Tataru, Strichartz estimates and local smoothing estimates for asymptotically flat Schrödinger equations,, \emph{J. Funct. Analysis}, 255 (2008), 1497. doi: 10.1016/j.jfa.2008.05.022.

[23]

H. Mizutani, Strichartz estimates for Schrödinger equations on scattering manifolds,, \emph{Comm. Partial Differential Equations}, 37 (2012), 169. doi: 10.1080/03605302.2011.593017.

[24]

H. Mizutani, Strichartz estimates for Schrödinger equations with variable coefficients and potentials at most linear at spatial infinity,, \emph{J. Math. Soc. Japan}, 65 (2013), 687.

[25]

H. Mizutani, Strichartz estimates for Schrödinger equations with variable coefficients and unbounded potentials,, \emph{Anal. PDE}, 6 (2013), 1857. doi: 10.2140/apde.2013.6.1857.

[26]

L. Robbiano and C. Zuily, Strichartz estimates for Schrödinger equations with variable coefficients,, \emph{M\'em. SMF. Math. Fr. (N.S.)}, 101-102 (2005), 101.

[27]

L. Robbiano and C. Zuily, Remark on the Kato smoothing effect for Schrödinger equation with superquadratic potentials,, \emph{Comm. Partial Differential Equations}, 33 (2008), 718. doi: 10.1080/03605300701517861.

[28]

D. Robert, Autour de lfapproximation semi-classique,, Progr. Math. 68 Birkh\, (1987).

[29]

I. Rodnianski and W. Schlag, Time decay for solutions of Schroödinger equations with rough and time dependent potentials,, \emph{Invent. Math.}, 155 (2004), 451. doi: 10.1007/s00222-003-0325-4.

[30]

W. Schlag, Dispersive estimates for Schrödinger operators: a survey,, in \emph{Mathematical aspects of nonlinear dispersive equations} (eds. B. J. et al.), 163 (2007), 255.

[31]

B. Simon, Schrödinger semigroups,, \emph{Bull. Amer. Math. Soc. (N.S.)}, 7 (1982), 447. doi: 10.1090/S0273-0979-1982-15041-8.

[32]

C. D. Sogge, Fourier Integrals in Classical Analysis,, Cambridge Tracts in Mathematics, (1993). doi: 10.1017/CBO9780511530029.

[33]

G. Staffilani and D. Tataru, Strichartz estimates for a Schrödinger operator with non smooth coefficients,, \emph{Comm. Partial Differential Equations}, 27 (2002), 1337. doi: 10.1081/PDE-120005841.

[34]

E. M. Stein and G. Weiss, Interpolation of operators with change of measures,, {Trans. Amer. Math. Soc.}, (1958), 159.

[35]

R. Strichartz, Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations,, \emph{Duke Math. J}, 44 (1977), 705.

[36]

D. Tataru, Parametrices and dispersive estimates for Schrödinger operators with variable coefficients,, \emph{Amer. J. Math.}, 130 (2008), 571. doi: 10.1353/ajm.0.0000.

[37]

K. Yajima, Existence of solutions for Schrödinger evolution equations,, \emph{Comm Math. Phys.}, 110 (1987), 415.

[38]

K. Yajima, Schrödinger evolution equation with magnetic fields,, \emph{J. d'Anal. Math.}, 56 (1991), 29. doi: 10.1007/BF02820459.

[39]

K. Yajima, Dispersive estimates for Schrödinger equations with threshold resonance and eigenvalue,, \emph{Comm. Math. Phys.}, 259 (2005), 475. doi: 10.1007/s00220-005-1375-9.

[40]

K. Yajima and G. Zhang, Local smoothing property and Strichartz inequality for Schrödinger equations with potentials superquadratic at infinity,, \emph{Comm. Partial Differential Equations}, 27 (2002), 1337.

[41]

S. Zheng, Littlewood-Paley theorem for Schrödinger operators,, \emph{Anal. Theory Appl.}, 22 (2006), 353. doi: 10.1007/s10496-006-0353-1.

show all references

References:
[1]

J. -M. Bouclet, Strichartz estimates on asymptotically hyperbolic manifolds,, \emph{Anal. PDE.}, 4 (2011), 1. doi: 10.2140/apde.2011.4.1.

[2]

J. -M. Bouclet and N. Tzvetkov, Strichartz estimates for long range perturbations,, \emph{Amer. J. Math.}, 129 (2007), 1565. doi: 10.1353/ajm.2007.0039.

[3]

J. -M. Bouclet and N. Tzvetkov, On global Strichartz estimates for non trapping metrics,, \emph{J. Funct. Analysis}, 254 (2008), 1661. doi: 10.1016/j.jfa.2007.11.018.

[4]

N. Burq, P. Gérard and N. Tzvetkov, Strichartz inequalities and the nonlinear Schrödinger equation on compact manifolds,, \emph{Amer. J. Math.}, 126 (2004), 569.

[5]

N. Burq, C. Guillarmou and A. Hassell, Strichartz estimates without loss on manifolds with hyperbolic trapped geodesics,, \emph{Geom. Funct. Anal.}, 20 (2010), 627. doi: 10.1007/s00039-010-0076-5.

[6]

N. Burq, F. Planchon, J. Stalker and A. S. Tahvildar-Zadeh, Strichartz estimates for the wave and Schrödinger equations with potentials of critical decay,, \emph{Indiana Univ. Math. J.}, 53 (2004), 1665. doi: 10.1512/iumj.2004.53.2541.

[7]

T. Cazenave, Semilinear Schrödinger Equations,, Courant. Lect. Nates Math. vol. 10, (2003).

[8]

P. D'Ancona and L. Fanelli, Smoothing estimates for the Schrödinger equation with unbounded potentials,, \emph{J. Differential Equations}, 246 (2009), 4552. doi: 10.1016/j.jde.2009.03.026.

[9]

P. D'Ancona, L. Fanelli, L. Vega and N. Visciglia, Endpoint Strichartz estimates for the magnetic Schrödinger equation,, \emph{J. Funct. Analysis}, 258 (2010), 3227. doi: 10.1016/j.jfa.2010.02.007.

[10]

S. Doi, Smoothness of solutions for Schrödinger equations with unbounded potentials,, \emph{Publ. Res. Inst. Math. Sci.}, 41 (2005), 175.

[11]

D. Dunford and J. T. Schwartz, Linear operators. Part II. Spectral theory. Selfadjoint operators in Hilbert space. With the assistance of William G. Bade and Robert G. Bartle,, Interscience Publishers John Wiley & Sons New York-London, (1963).

[12]

L. Fanelli, V. Felli, M. Fontelos and M. A. Primo, Time decay of scaling critical electromagnetic Schrödinger flows,, \emph{Commun. Math. Phs.}, 324 (2013), 1033. doi: 10.1007/s00220-013-1830-y.

[13]

D. Fujiwara, Remarks on convergence of the Feynman path integrals,, \emph{Duke Math. J.}, 47 (1980), 559.

[14]

J. Ginibre and G. Velo, The global Cauchy problem for the non linear Schrödinger equation revisited,, \emph{Ann. lHP-Analyse non lin\'eaire.}, 2 (1985), 309.

[15]

A. Hassell, T. Tao and J. Wunsch, Sharp Strichartz estimates on nontrapping asymptotically conic manifolds,, \emph{Amer. J. Math.}, 128 (2006), 963.

[16]

B. Helffer and J. Sjöstrand, Equation de Schrödinger avec champ magnétique et équation de Harper,, in \emph{Schr\, 345 (1989), 118. doi: 10.1007/3-540-51783-9_19.

[17]

L. Hörmander, The Analysis of Linear Partial Differential Operators III,, Springer- Verlag, (1985).

[18]

A. Iwatsuka, Essential self-adjointness of the Schrödinger operators with magnetic fields diverging at infinity,, \emph{Publ. Res. Inst. Math. Sci.}, 26 (1990), 841. doi: 10.2977/prims/1195170737.

[19]

J.- L. Journé, A. Soffer, C. D. Sogge, Decay estimates for Schrödinger operators,, \emph{Comm. Pure Appl. Math.}, (1991), 573. doi: 10.1002/cpa.3160440504.

[20]

M. Keel and T. Tao, Endpoint Strichartz Estimates,, \emph{Amer. J. Math.}, 120 (1998), 955.

[21]

A. Martinez, An Introduction to Semiclassical and Microlocal Analysis,, Springer-Verlag, (2002). doi: 10.1007/978-1-4757-4495-8.

[22]

J. Marzuola, J. Metcalfe and D. Tataru, Strichartz estimates and local smoothing estimates for asymptotically flat Schrödinger equations,, \emph{J. Funct. Analysis}, 255 (2008), 1497. doi: 10.1016/j.jfa.2008.05.022.

[23]

H. Mizutani, Strichartz estimates for Schrödinger equations on scattering manifolds,, \emph{Comm. Partial Differential Equations}, 37 (2012), 169. doi: 10.1080/03605302.2011.593017.

[24]

H. Mizutani, Strichartz estimates for Schrödinger equations with variable coefficients and potentials at most linear at spatial infinity,, \emph{J. Math. Soc. Japan}, 65 (2013), 687.

[25]

H. Mizutani, Strichartz estimates for Schrödinger equations with variable coefficients and unbounded potentials,, \emph{Anal. PDE}, 6 (2013), 1857. doi: 10.2140/apde.2013.6.1857.

[26]

L. Robbiano and C. Zuily, Strichartz estimates for Schrödinger equations with variable coefficients,, \emph{M\'em. SMF. Math. Fr. (N.S.)}, 101-102 (2005), 101.

[27]

L. Robbiano and C. Zuily, Remark on the Kato smoothing effect for Schrödinger equation with superquadratic potentials,, \emph{Comm. Partial Differential Equations}, 33 (2008), 718. doi: 10.1080/03605300701517861.

[28]

D. Robert, Autour de lfapproximation semi-classique,, Progr. Math. 68 Birkh\, (1987).

[29]

I. Rodnianski and W. Schlag, Time decay for solutions of Schroödinger equations with rough and time dependent potentials,, \emph{Invent. Math.}, 155 (2004), 451. doi: 10.1007/s00222-003-0325-4.

[30]

W. Schlag, Dispersive estimates for Schrödinger operators: a survey,, in \emph{Mathematical aspects of nonlinear dispersive equations} (eds. B. J. et al.), 163 (2007), 255.

[31]

B. Simon, Schrödinger semigroups,, \emph{Bull. Amer. Math. Soc. (N.S.)}, 7 (1982), 447. doi: 10.1090/S0273-0979-1982-15041-8.

[32]

C. D. Sogge, Fourier Integrals in Classical Analysis,, Cambridge Tracts in Mathematics, (1993). doi: 10.1017/CBO9780511530029.

[33]

G. Staffilani and D. Tataru, Strichartz estimates for a Schrödinger operator with non smooth coefficients,, \emph{Comm. Partial Differential Equations}, 27 (2002), 1337. doi: 10.1081/PDE-120005841.

[34]

E. M. Stein and G. Weiss, Interpolation of operators with change of measures,, {Trans. Amer. Math. Soc.}, (1958), 159.

[35]

R. Strichartz, Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations,, \emph{Duke Math. J}, 44 (1977), 705.

[36]

D. Tataru, Parametrices and dispersive estimates for Schrödinger operators with variable coefficients,, \emph{Amer. J. Math.}, 130 (2008), 571. doi: 10.1353/ajm.0.0000.

[37]

K. Yajima, Existence of solutions for Schrödinger evolution equations,, \emph{Comm Math. Phys.}, 110 (1987), 415.

[38]

K. Yajima, Schrödinger evolution equation with magnetic fields,, \emph{J. d'Anal. Math.}, 56 (1991), 29. doi: 10.1007/BF02820459.

[39]

K. Yajima, Dispersive estimates for Schrödinger equations with threshold resonance and eigenvalue,, \emph{Comm. Math. Phys.}, 259 (2005), 475. doi: 10.1007/s00220-005-1375-9.

[40]

K. Yajima and G. Zhang, Local smoothing property and Strichartz inequality for Schrödinger equations with potentials superquadratic at infinity,, \emph{Comm. Partial Differential Equations}, 27 (2002), 1337.

[41]

S. Zheng, Littlewood-Paley theorem for Schrödinger operators,, \emph{Anal. Theory Appl.}, 22 (2006), 353. doi: 10.1007/s10496-006-0353-1.

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