Bounds on the growth of high Sobolev norms of solutions to 2D Hartree equations

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October 2012, 32(10): 3733-3771. doi: 10.3934/dcds.2012.32.3733

Bounds on the growth of high Sobolev norms of solutions to 2D Hartree equations

Vedran Sohinger ^1,

1.	Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, United States

Received May 2010 Revised January 2012 Published May 2012

Abstract
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In this paper, we consider Hartree-type equations on the two-dimensional torus and on the plane. We prove polynomial bounds on the growth of high Sobolev norms of solutions to these equations. The proofs of our results are based on the adaptation to two dimensions of the techniques we had previously used in [49, 50] to study the analogous problem in one dimension. Since we are working in two dimensions, a more detailed analysis of the resonant frequencies is needed, as was previously used in the work of Colliander-Keel-Staffilani-Takaoka-Tao [19].

Keywords: growth of high Sobolev norms, resonant decomposition., Hartree equation, nonlinear Schrödinger equation.

Mathematics Subject Classification: 35Q5.

Citation: Vedran Sohinger. Bounds on the growth of high Sobolev norms of solutions to 2D Hartree equations. Discrete and Continuous Dynamical Systems, 2012, 32 (10) : 3733-3771. doi: 10.3934/dcds.2012.32.3733

References:

[1]	D. Benney and A. Newell, Random wave closures, Stud. Appl. Math., 48 (1969), 29-53.
[2]	D. Benney and P. Saffman, Nonlinear interactions of random waves in a dispersive medium, Proc. Roy. Soc. A, 289 (1966), 301-320. doi: 10.1098/rspa.1966.0013.
[3]	J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I. Schrödinger equations, Geom. Funct. Anal., 3 (1993), 107-156.
[4]	J. Bourgain, On the growth in time of higher Sobolev norms of smooth solutions of Hamiltonian PDE, Int. Math. Research Notices, 1996, 277-304.
[5]	J. Bourgain, Refinements of Strichartz's inequality and applications to 2D-NLS with critical nonlinearity, Int. Math. Research Notices, 1998, 253-283.
[6]	J. Bourgain, "Nonlinear Schrödinger Equations," in "Hyperbolic Equations and Frequency Interactions'' (eds. L. Caffarelli and W. E), IAS/Park City Mathematics Series, 5, AMS, Providence, RI, (1999), 3-157.
[7]	J. Bourgain, Global solutions of nonlinear Schrödinger equations, AMS Colloquium Publications, 46, AMS, Providence, RI, 1999.
[8]	J. Bourgain, On growth of Sobolev norms in linear Schrödinger equations with smooth time dependent potential, J. Anal. Math., 77 (1999), 315-348. doi: 10.1007/BF02791265.
[9]	N. Burq, P. Gérard and N. Tzvetkov, An instability property of the nonlinear Schrödinger equation on $S^d$, Mathematical Research Letters, 9 (2002), 323-335.
[10]	N. Burq, P. Gérard and N. Tzvetkov, Bilinear eigenfunction estimates and the nonlinear Schrödinger equation on surfaces, Invent. Math., 159 (2005), 187-223. doi: 10.1007/s00222-004-0388-x.
[11]	F. Catoire and W.-M. Wang, Bounds on Sobolev norms for the nonlinear Schrödinger equation on general tori, preprint, (2008), arXiv:0809.4633.
[12]	T. Cazenave, "Semilinear Schrödinger Equations," Courant Lecture Notes in Mathematics, 10, New York University, CIMS, New York, AMS, Providence, RI, 2003.
[13]	J. Colliander, J.-M. Delort, C. E. Kenig and G. Staffilani, Bilinear estimates and applications to 2D NLS, Trans. of the American Math. Soc., 353 (2001), 3307-3325. doi: 10.1090/S0002-9947-01-02760-X.
[14]	J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Global well-posedness for Schrödinger equations with derivative, SIAM J. Math. Anal., 33 (2001), 649-669. doi: 10.1137/S0036141001384387.
[15]	J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Polynomial upper bounds for the orbital instability of the 1D cubic NLS below the energy norm, Discrete Contin. Dyn. Syst., 9 (2003), 31-54.
[16]	J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Sharp global well-posedness for KdV and modified KdV on $\mathbb{R}$ and $\mathbb{T}$, J. Amer. Math. Soc., 16 (2003), 705-749 (electronic). doi: 10.1090/S0894-0347-03-00421-1.
[17]	J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Multilinear estimates for periodic KdV equations, and applications, J. Funct. Anal., 211 (2004), 173-218. doi: 10.1016/S0022-1236(03)00218-0.
[18]	J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, A refined global well-posedness for Schrödinger equations with derivative, SIAM J. Math. Anal., 34 (2002), 64-86. doi: 10.1137/S0036141001394541.
[19]	J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Resonant decompositions and the I-method for cubic nonlinear Schrödinger equation on $\mathbb{R}^2$, Disc. and Cont. Dynam. Sys., 21 (2008), 665-686. doi: 10.3934/dcds.2008.21.665.
[20]	J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in $\mathbb{R}^3$, Ann. of Math. (2), 167 (2008), 767-865. doi: 10.4007/annals.2008.167.767.
[21]	J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Transfer of energy to high frequencies in the cubic nonlinear Schrödinger equation, Invent. Math., 181 (2010), 39-113. doi: 10.1007/s00222-010-0242-2.
[22]	J.-M. Delort, Growth of Sobolev norms of solutions of linear Schrödinger equations on some compact manifolds, preprint, International Mathematics Research Notices, 2010, 2305-2328.
[23]	B. Dodson, Global well-posedness and scattering for the defocusing, $L^2 $- critical, nonlinear Schrödinger equation when $d \geq 3$, preprint, (2009), arXiv:0912.2467.
[24]	B. Dodson, Global well-posedness and scattering for the defocusing, $L^2 $- critical, nonlinear Schrödinger equation when $d=2$, preprint, (2010), arXiv:1006.1365.
[25]	B. Dodson, Global well-posedness and scattering for the defocusing, $L^2 $- critical, nonlinear Schrödinger equation when $d=1$, preprint, (2009), arXiv:1010.0040.
[26]	J. Duoandikoetxea, "Fourier Analysis," Graduate Studies in Mathematics, 29, AMS, Providence, RI, 2001.
[27]	J. Fröhlich and E. Lenzmann, "Mean-Field Limit of Quantum Bose Gases and Nonlinear Hartree Equation," Séminaire: É.D.P. 2003-2004, Exposé No. XIX, Sémin. Équ. Dériv. Partielles, École Polytech., Palaiseau, (2004), 26 pp.
[28]	J. Ginibre and T. Ozawa, Long-range scattering for non-linear Schrödinger and Hartree equations in space dimension $n\geq 2$, Comm. Math. Phys., 151 (1993), 619-645. doi: 10.1007/BF02097031.
[29]	J. Ginibre and G. Velo, On a class of nonlinear Schrödinger equations. I. The Cauchy problem, general case, J. Funct. Anal., 32 (1979), 1-32. doi: 10.1016/0022-1236(79)90076-4.
[30]	J. Ginibre and G. Velo, On a class of nonlinear Schrödinger equations with nonlocal interaction, Math. Z., 170 (1980), 109-136. doi: 10.1007/BF01214768.
[31]	J. Ginibre and G. Velo, Scattering theory in the energy space for a class of Hartree equations, Rev. Math. Phys., 12 (2000), 361-429. doi: 10.1142/S0129055X00000137.
[32]	J. Ginibre and G. Velo, Long range scattering and modified wave operators for some Hartree type equations. II, Ann. H. P., 1 (2000), 753-800.
[33]	J. Ginibre and G. Velo, Long range scattering and modified wave operators for some Hartree type equations. III. Gevrey spaces and low dimensions, J. Diff. Eq., 175 (2001), 415-501. doi: 10.1006/jdeq.2000.3969.
[34]	A. Grünrock, On the Cauchy- and periodic boundary value problem for a certain class of derivative nonlinear Schrödinger equations, preprint, 2006.
[35]	, Z. Hani, Private communication.
[36]	N. Hayashi, P. Naumkin and T. Ozawa, "Scattering Theory for the Hartree Equation," Hokkaido University Preprints, Series 358, Nov., 1996.
[37]	C. Kenig, G. Ponce and L. Vega, The Cauchy problem for the Korteweg-de Vries equation in Sobolev spaces of negative indices, Duke Math. J., 71 (1993), 1-21.
[38]	C. Kenig, G. Ponce and L. Vega, Quadratic forms for the 1-D semilinear Schrödinger equation, Transactions of the AMS, 348 (1996), 3323-3353. doi: 10.1090/S0002-9947-96-01645-5.
[39]	C. Miao, Y. Wu and G. Xu, Dynamics for the focusing, energy-critical nonlinear Hartree equation, preprint, (2011), arXiv:1104.1229.
[40]	C. Miao, G. Xu and L. Zhao, Global well-posedness and scattering for the energy critical, defocusing Hartree equation for radial data, J. Funct. Anal., 253 (2007), 605-627. doi: 10.1016/j.jfa.2007.09.008.
[41]	C. Miao, G. Xu and L. Zhao, The Cauchy problem for the Hartree equation, J. PDEs, 21 (2008), 22-24.
[42]	C. Miao, G. Xu and L. Zhao, Global well-posedness, scattering, and blow-up for the energy critical, focusing Hartree equation in the radial case, Coll. Math., 114 (2009), 213-236. doi: 10.4064/cm114-2-5.
[43]	C. Miao, G. Xu and L. Zhao, Global well-posedness and scattering for the defocusing $H^{\frac{1}{2}}$ -subcritical Hartree equation on $\mathbbmathbb{R}^{d}$, Ann. I. H. Poincaré, NA, 26 (2009), 1831-1852.
[44]	C. Miao, G. Xu and L. Zhao, Global well-posedness and scattering for the energy-critical, defocusing Hartree equation in $\mathbbR^{1+n}$, Comm. PDEs, 36 (2011), 729-776. doi: 10.1080/03605302.2010.531073.
[45]	C. Morawetz and W. A. Strauss, Decay and scattering of solutions of a nonlinear relativistic wave equation, Comm. Pure. Appl. Math., 25 (1972), 1-31. doi: 10.1002/cpa.3160250103.
[46]	B. Schlein, "Derivation of Effective Evolution Equations from Microscopic Quantum Dynamics," Lecture Notes, Clay Summer School on Evolution Equations, Zurich, (2008), arXiv:0807.4307.
[47]	C. Sogge, Oscillatory integrals and spherical harmonics, Duke Math. Jour., 53 (1986), 43-65. doi: 10.1215/S0012-7094-86-05303-2.
[48]	C. Sogge, Concerning the $\ell^p$ norm of spectral clusters for second order elliptic operators on compact manifolds, Jour. of Funct. Anal., 77 (1988), 123-138. doi: 10.1016/0022-1236(88)90081-X.
[49]	V. Sohinger, Bounds on the growth of high Sobolev norms of solutions to Nonlinear Schrödinger Equations on $S^1$, to appear in Diff. and Int. Eqs., (2010), arXiv:1003.5705.
[50]	V. Sohinger, Bounds on the growth of high Sobolev norms of solutions to nonlinear Schrödinger equations on $\mathbb{R}$, to appear in Indiana Univ. Math. J., (2010), arXiv:1003.5707.
[51]	G. Staffilani, On the growth of high Sobolev norms of solutions for KdV and Schrödinger equations, Duke Math. J., 86 (1997), 109-142. doi: 10.1215/S0012-7094-97-08604-X.
[52]	G. Staffilani, Quadratic forms for a 2-D semilinear Schrödinger equation, Duke Math. J., 86 (1997), 79-107. doi: 10.1215/S0012-7094-97-08603-8.
[53]	T. Tao, "Nonlinear Dispersive Equations: Local and Global Analysis," CBMS Reg. Conf. Series in Math., 106, Published for the Conference Board of the Mathematical Sciences, Washington, DC, AMS, Providence, RI, 2006.
[54]	V. E. Zakharov, Stability of periodic waves of finite amplitude on a surface of deep fluid, J. Appl. Mech. Tech. Phys., 9 (1968), 190-194.
[55]	S.-J. Zhong, The growth in time of higher Sobolev norms of solutions to Schrödinger equations on compact Riemannian manifolds, J. Differential Equations, 245 (2008), 359-376. doi: 10.1016/j.jde.2008.03.008.

show all references

References:

[1]	D. Benney and A. Newell, Random wave closures, Stud. Appl. Math., 48 (1969), 29-53.
[2]	D. Benney and P. Saffman, Nonlinear interactions of random waves in a dispersive medium, Proc. Roy. Soc. A, 289 (1966), 301-320. doi: 10.1098/rspa.1966.0013.
[3]	J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I. Schrödinger equations, Geom. Funct. Anal., 3 (1993), 107-156.
[4]	J. Bourgain, On the growth in time of higher Sobolev norms of smooth solutions of Hamiltonian PDE, Int. Math. Research Notices, 1996, 277-304.
[5]	J. Bourgain, Refinements of Strichartz's inequality and applications to 2D-NLS with critical nonlinearity, Int. Math. Research Notices, 1998, 253-283.
[6]	J. Bourgain, "Nonlinear Schrödinger Equations," in "Hyperbolic Equations and Frequency Interactions'' (eds. L. Caffarelli and W. E), IAS/Park City Mathematics Series, 5, AMS, Providence, RI, (1999), 3-157.
[7]	J. Bourgain, Global solutions of nonlinear Schrödinger equations, AMS Colloquium Publications, 46, AMS, Providence, RI, 1999.
[8]	J. Bourgain, On growth of Sobolev norms in linear Schrödinger equations with smooth time dependent potential, J. Anal. Math., 77 (1999), 315-348. doi: 10.1007/BF02791265.
[9]	N. Burq, P. Gérard and N. Tzvetkov, An instability property of the nonlinear Schrödinger equation on $S^d$, Mathematical Research Letters, 9 (2002), 323-335.
[10]	N. Burq, P. Gérard and N. Tzvetkov, Bilinear eigenfunction estimates and the nonlinear Schrödinger equation on surfaces, Invent. Math., 159 (2005), 187-223. doi: 10.1007/s00222-004-0388-x.
[11]	F. Catoire and W.-M. Wang, Bounds on Sobolev norms for the nonlinear Schrödinger equation on general tori, preprint, (2008), arXiv:0809.4633.
[12]	T. Cazenave, "Semilinear Schrödinger Equations," Courant Lecture Notes in Mathematics, 10, New York University, CIMS, New York, AMS, Providence, RI, 2003.
[13]	J. Colliander, J.-M. Delort, C. E. Kenig and G. Staffilani, Bilinear estimates and applications to 2D NLS, Trans. of the American Math. Soc., 353 (2001), 3307-3325. doi: 10.1090/S0002-9947-01-02760-X.
[14]	J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Global well-posedness for Schrödinger equations with derivative, SIAM J. Math. Anal., 33 (2001), 649-669. doi: 10.1137/S0036141001384387.
[15]	J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Polynomial upper bounds for the orbital instability of the 1D cubic NLS below the energy norm, Discrete Contin. Dyn. Syst., 9 (2003), 31-54.
[16]	J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Sharp global well-posedness for KdV and modified KdV on $\mathbb{R}$ and $\mathbb{T}$, J. Amer. Math. Soc., 16 (2003), 705-749 (electronic). doi: 10.1090/S0894-0347-03-00421-1.
[17]	J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Multilinear estimates for periodic KdV equations, and applications, J. Funct. Anal., 211 (2004), 173-218. doi: 10.1016/S0022-1236(03)00218-0.
[18]	J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, A refined global well-posedness for Schrödinger equations with derivative, SIAM J. Math. Anal., 34 (2002), 64-86. doi: 10.1137/S0036141001394541.
[19]	J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Resonant decompositions and the I-method for cubic nonlinear Schrödinger equation on $\mathbb{R}^2$, Disc. and Cont. Dynam. Sys., 21 (2008), 665-686. doi: 10.3934/dcds.2008.21.665.
[20]	J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in $\mathbb{R}^3$, Ann. of Math. (2), 167 (2008), 767-865. doi: 10.4007/annals.2008.167.767.
[21]	J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Transfer of energy to high frequencies in the cubic nonlinear Schrödinger equation, Invent. Math., 181 (2010), 39-113. doi: 10.1007/s00222-010-0242-2.
[22]	J.-M. Delort, Growth of Sobolev norms of solutions of linear Schrödinger equations on some compact manifolds, preprint, International Mathematics Research Notices, 2010, 2305-2328.
[23]	B. Dodson, Global well-posedness and scattering for the defocusing, $L^2 $- critical, nonlinear Schrödinger equation when $d \geq 3$, preprint, (2009), arXiv:0912.2467.
[24]	B. Dodson, Global well-posedness and scattering for the defocusing, $L^2 $- critical, nonlinear Schrödinger equation when $d=2$, preprint, (2010), arXiv:1006.1365.
[25]	B. Dodson, Global well-posedness and scattering for the defocusing, $L^2 $- critical, nonlinear Schrödinger equation when $d=1$, preprint, (2009), arXiv:1010.0040.
[26]	J. Duoandikoetxea, "Fourier Analysis," Graduate Studies in Mathematics, 29, AMS, Providence, RI, 2001.
[27]	J. Fröhlich and E. Lenzmann, "Mean-Field Limit of Quantum Bose Gases and Nonlinear Hartree Equation," Séminaire: É.D.P. 2003-2004, Exposé No. XIX, Sémin. Équ. Dériv. Partielles, École Polytech., Palaiseau, (2004), 26 pp.
[28]	J. Ginibre and T. Ozawa, Long-range scattering for non-linear Schrödinger and Hartree equations in space dimension $n\geq 2$, Comm. Math. Phys., 151 (1993), 619-645. doi: 10.1007/BF02097031.
[29]	J. Ginibre and G. Velo, On a class of nonlinear Schrödinger equations. I. The Cauchy problem, general case, J. Funct. Anal., 32 (1979), 1-32. doi: 10.1016/0022-1236(79)90076-4.
[30]	J. Ginibre and G. Velo, On a class of nonlinear Schrödinger equations with nonlocal interaction, Math. Z., 170 (1980), 109-136. doi: 10.1007/BF01214768.
[31]	J. Ginibre and G. Velo, Scattering theory in the energy space for a class of Hartree equations, Rev. Math. Phys., 12 (2000), 361-429. doi: 10.1142/S0129055X00000137.
[32]	J. Ginibre and G. Velo, Long range scattering and modified wave operators for some Hartree type equations. II, Ann. H. P., 1 (2000), 753-800.
[33]	J. Ginibre and G. Velo, Long range scattering and modified wave operators for some Hartree type equations. III. Gevrey spaces and low dimensions, J. Diff. Eq., 175 (2001), 415-501. doi: 10.1006/jdeq.2000.3969.
[34]	A. Grünrock, On the Cauchy- and periodic boundary value problem for a certain class of derivative nonlinear Schrödinger equations, preprint, 2006.
[35]	, Z. Hani, Private communication.
[36]	N. Hayashi, P. Naumkin and T. Ozawa, "Scattering Theory for the Hartree Equation," Hokkaido University Preprints, Series 358, Nov., 1996.
[37]	C. Kenig, G. Ponce and L. Vega, The Cauchy problem for the Korteweg-de Vries equation in Sobolev spaces of negative indices, Duke Math. J., 71 (1993), 1-21.
[38]	C. Kenig, G. Ponce and L. Vega, Quadratic forms for the 1-D semilinear Schrödinger equation, Transactions of the AMS, 348 (1996), 3323-3353. doi: 10.1090/S0002-9947-96-01645-5.
[39]	C. Miao, Y. Wu and G. Xu, Dynamics for the focusing, energy-critical nonlinear Hartree equation, preprint, (2011), arXiv:1104.1229.
[40]	C. Miao, G. Xu and L. Zhao, Global well-posedness and scattering for the energy critical, defocusing Hartree equation for radial data, J. Funct. Anal., 253 (2007), 605-627. doi: 10.1016/j.jfa.2007.09.008.
[41]	C. Miao, G. Xu and L. Zhao, The Cauchy problem for the Hartree equation, J. PDEs, 21 (2008), 22-24.
[42]	C. Miao, G. Xu and L. Zhao, Global well-posedness, scattering, and blow-up for the energy critical, focusing Hartree equation in the radial case, Coll. Math., 114 (2009), 213-236. doi: 10.4064/cm114-2-5.
[43]	C. Miao, G. Xu and L. Zhao, Global well-posedness and scattering for the defocusing $H^{\frac{1}{2}}$ -subcritical Hartree equation on $\mathbbmathbb{R}^{d}$, Ann. I. H. Poincaré, NA, 26 (2009), 1831-1852.
[44]	C. Miao, G. Xu and L. Zhao, Global well-posedness and scattering for the energy-critical, defocusing Hartree equation in $\mathbbR^{1+n}$, Comm. PDEs, 36 (2011), 729-776. doi: 10.1080/03605302.2010.531073.
[45]	C. Morawetz and W. A. Strauss, Decay and scattering of solutions of a nonlinear relativistic wave equation, Comm. Pure. Appl. Math., 25 (1972), 1-31. doi: 10.1002/cpa.3160250103.
[46]	B. Schlein, "Derivation of Effective Evolution Equations from Microscopic Quantum Dynamics," Lecture Notes, Clay Summer School on Evolution Equations, Zurich, (2008), arXiv:0807.4307.
[47]	C. Sogge, Oscillatory integrals and spherical harmonics, Duke Math. Jour., 53 (1986), 43-65. doi: 10.1215/S0012-7094-86-05303-2.
[48]	C. Sogge, Concerning the $\ell^p$ norm of spectral clusters for second order elliptic operators on compact manifolds, Jour. of Funct. Anal., 77 (1988), 123-138. doi: 10.1016/0022-1236(88)90081-X.
[49]	V. Sohinger, Bounds on the growth of high Sobolev norms of solutions to Nonlinear Schrödinger Equations on $S^1$, to appear in Diff. and Int. Eqs., (2010), arXiv:1003.5705.
[50]	V. Sohinger, Bounds on the growth of high Sobolev norms of solutions to nonlinear Schrödinger equations on $\mathbb{R}$, to appear in Indiana Univ. Math. J., (2010), arXiv:1003.5707.
[51]	G. Staffilani, On the growth of high Sobolev norms of solutions for KdV and Schrödinger equations, Duke Math. J., 86 (1997), 109-142. doi: 10.1215/S0012-7094-97-08604-X.
[52]	G. Staffilani, Quadratic forms for a 2-D semilinear Schrödinger equation, Duke Math. J., 86 (1997), 79-107. doi: 10.1215/S0012-7094-97-08603-8.
[53]	T. Tao, "Nonlinear Dispersive Equations: Local and Global Analysis," CBMS Reg. Conf. Series in Math., 106, Published for the Conference Board of the Mathematical Sciences, Washington, DC, AMS, Providence, RI, 2006.
[54]	V. E. Zakharov, Stability of periodic waves of finite amplitude on a surface of deep fluid, J. Appl. Mech. Tech. Phys., 9 (1968), 190-194.
[55]	S.-J. Zhong, The growth in time of higher Sobolev norms of solutions to Schrödinger equations on compact Riemannian manifolds, J. Differential Equations, 245 (2008), 359-376. doi: 10.1016/j.jde.2008.03.008.

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