June  2017, 37(6): 2999-3023. doi: 10.3934/dcds.2017129

Scattering of solutions to the nonlinear Schrödinger equations with regular potentials

1. 

College of Science, Hohai University, Nanjing 210098, Jiangsu, China

2. 

Wu Wen-Tsun Key Laboratory of Mathematics, Chinese Academy of Sciences and Department of Mathematic, University of Science and Technology of China, Hefei 230026, Anhui, China

* Corresponding authorr: Lifeng Zhao

Received  November 2016 Revised  January 2017 Published  February 2017

Fund Project: X. Cheng has been partially supported by the NSF grant of China (No. 11526072) and "the Fundamental Research Funds for the Central Universities"(No. 2014B14214). L. Zhao has been partially supported by the NSFC grant of China (No. 10901148, No. 11371337) and by Youth Innovation Promotion Association CAS.

In this paper, we prove the scattering of radial solutions to high dimensional energy-critical nonlinear Schrödinger equations with regular potentials in the defocusing case.

Citation: Xing Cheng, Ze Li, Lifeng Zhao. Scattering of solutions to the nonlinear Schrödinger equations with regular potentials. Discrete and Continuous Dynamical Systems, 2017, 37 (6) : 2999-3023. doi: 10.3934/dcds.2017129
References:
[1]

P. Alsholm and G. Schmidt, Spectral and scattering theory for Schrödinger operators, Arch. Rational Mech. Anal, 40 (1971), 281-311.  doi: 10.1007/BF00252679.

[2]

P. AntonelliR. Carles and J. D. Silva, Scattering for nonlinear Schrödinger equation under partial harmonic confinement, Comm. Math. Phys., 334 (2015), 367-396.  doi: 10.1007/s00220-014-2166-y.

[3]

V. Banica and N. Visciglia, Scattering for non linear Schrödinger equation with a delta potential, J. Differential Equations, 260 (2016), 4410-4439.  doi: 10.1016/j.jde.2015.11.016.

[4]

M. Beceanu and M. Goldberg, Schrödinger dispersive estimates for a scaling-critical class of potentials, Comm. Math. Phys., 314 (2012), 471-481.  doi: 10.1007/s00220-012-1435-x.

[5]

J. Bourgain, Global wellposedness of defocusing critical nonlinear Schrödinger equation in the radial case, J. Amer. Math. Soc., 12 (1999), 145-171.  doi: 10.1090/S0894-0347-99-00283-0.

[6]

P. Chen, J. Magniez and E. M. Ouhabaz, Riesz transforms on non-compact manifolds, arXiv: 1411.0137.

[7]

J. CollianderM. Czubak and J. Lee, Interaction Morawetz estimate for the magnetic Schrödinger equation and applications, Adv. Differential Equations, 19 (2014), 805-832. 

[8]

J. CollianderM. KeelG. StaffilaniH. Takaoka and T. Tao, Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in $\mathbb{R}^3$, Ann. of Math., 167 (2008), 767-865.  doi: 10.4007/annals.2008.167.767.

[9]

S. CuccagnaV. Georgiev and N. Visciglia, Decay and scattering of small solutions of pure power NLS in $\mathbb{R}$ with p>3 and with a potential, Comm. Pure Appl. Math., 67 (2014), 957-981.  doi: 10.1002/cpa.21465.

[10]

P. D'anconaL. FanelliL. Vega and N. Visciglia, Endpoint Strichartz estimates for the magnetic Schrödinger equation, J. Funct. Anal., 258 (2010), 3227-3240.  doi: 10.1016/j.jfa.2010.02.007.

[11]

P. D'ancona and V. Pierfelice, On the wave equation with a large rough potential, J. Funct. Anal., 227 (2005), 30-77.  doi: 10.1016/j.jfa.2005.05.013.

[12]

B. Dodson, Global well-posedness and scattering for the defocusing, L2-critical, nonlinear Schrödinger equation when d ≥ 3, J. Amer. Math. Soc., 25 (2012), 429-463.  doi: 10.1090/S0894-0347-2011-00727-3.

[13]

B. Dodson, Global well-posedness and scattering for the defocusing, L2-critical, nonlinear Schrödinger equation when d = 2, Duke Math. J., 165 (2016), no. 18,3435-3516.  doi: 10.1090/S0894-0347-2011-00727-3.

[14]

B. Dodson, Global well-posedness and scattering for the defocusing, L2-critical, nonlinear Schrödinger equation when d = 1 to appear in Amer. J. Math. , arXiv: 1010.0040. doi: 10.1090/S0894-0347-2011-00727-3.

[15]

B. Dodson, Global well-posedness and scattering for the mass critical nonlinear Schrödinger equation with mass below the mass of the ground state, Advances in Mathematics, 285 (2015), 1589-1618.  doi: 10.1016/j.aim.2015.04.030.

[16]

B. Dodson, Global well-posedness and scattering for the focusing, energy-critical nonlinear Schrödinger problem in dimension d = 4 for initial data below a ground state threshold, arXiv: 1409. 1950.

[17]

J. GinibreT. Ozawa and G. Velo, On the existence of the wave operators for a class of nonlinear Schrödinger equations, Ann. Inst. H. Poincare Phys. Theor., 60 (1994), 211-239. 

[18]

J. Ginibre and G. Velo, Scattering theory in the energy space for a class of nonlinear Schrödinger equations, J. Math. Pures Appl.(9), 64 (1985), 363-401. 

[19]

R. H. GoodmanR. E. Slusher and M. I. Weinstein, Stopping light on a defect, J. Opt. Soc. Am. B, 19 (2002), 1635-1652.  doi: 10.1364/JOSAB.19.001635.

[20]

R. H. GoodmanM. I. Weinstein and P. J. Holmes, Nonlinear propagation of light in one-dimensional periodic structures, J. Nonlinear Sci., 11 (2001), 123-168.  doi: 10.1007/s00332-001-0002-y.

[21]

Z. Hani and L. Thomann, Asymptotic behavior of the nonlinear Schrödinger equation with harmonic trapping, Comm. Pure Appl. Math., 69 (2016), 1727-1776.  doi: 10.1002/cpa.21594.

[22]

K. Hepp, The classical limit for quantum mechanical correlation functions, Comm. Math. Phys., 35 (1974), 265-277.  doi: 10.1007/BF01646348.

[23]

Y. Hong, Scattering for a nonlinear Schrödinger equation with a potential, Comm. Pure Appl. Anal.(5), 15 (2016), 1571-1601.  doi: 10.3934/cpaa.2016003.

[24]

S. IbrahimN. Masmoudi and K. Nakanishi, Scattering threshold for the focusing nonlinear Klein-Gordon equation, Anal. PDE, 4 (2011), 405-460.  doi: 10.2140/apde.2011.4.405.

[25]

J. L. JourneA. Soffer and C. D. Sogge, Decay estimates for Schrödinger operators, Comm. Pure Appl. Math., 44 (1991), 573-604.  doi: 10.1002/cpa.3160440504.

[26]

M. Keel and T. Tao, Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955-980.  doi: 10.1353/ajm.1998.0039.

[27]

S. Keraani, On the defect of compactness for the Strichartz estimates for the Schrödinger equations, J. Differential Equations, 175 (2001), 353-392.  doi: 10.1006/jdeq.2000.3951.

[28]

C. E. Kenig and F. Merle, Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case, Invent. Math., 166 (2006), 645-675.  doi: 10.1007/s00222-006-0011-4.

[29]

R. Killip and M. Visan, The focusing energy-critical nonlinear Schrödinger equation in dimensions five and higher, Amer. J. Math., 132 (2010), 361-424.  doi: 10.1353/ajm.0.0107.

[30]

R. KillipM. Visan and X. Zhang, The mass-critical nonlinear Schrödinger equation with radial data in dimensions three and higher, Analysis and PDE, 1 (2009), 229-266.  doi: 10.2140/apde.2008.1.229.

[31]

D. Lafontaine, Scattering for NLS with a potential on the line, Asymptotic Analysis, 100 (2016), 21-39.  doi: 10.3233/ASY-161384.

[32]

E. H. LiebR. Seiringer and J. Yngvason, A rigorous derivation of the Gross-Pitaevskii energy functional for a two-dimensional Bose gas, Comm. Math. Phys., 224 (2001), 17-31.  doi: 10.1007/s002200100533.

[33]

H. P. McKean and J. Shatah, The nonlinear Schrödinger equation and the nonlinear heat equation reduction to linear form, Comm. Pure Appl. Math., 44 (1991), 1067-1080.  doi: 10.1002/cpa.3160440817.

[34]

K. Nakanishi, Energy scattering for nonlinear Klein-Gordon and Schrödinger equations in spatial dimensions 1 and 2, Journal of Functional Analysis, 169 (1999), 201-225.  doi: 10.1006/jfan.1999.3503.

[35]

F. Planchon and L. Vega, Bilinear virial identities and applications, Ann. Sci. Ec. Norm. Super., 42 (2009), 261-290. 

[36]

I. Rodnianski and W. Schlag, Time decay for solutions of Schrödinger equations with rough and time-dependent potentials, Invent. Math., 155 (2004), 451-513.  doi: 10.1007/s00222-003-0325-4.

[37]

E. Ryckman and M. Visan, Global well-posedness and scattering for the defocusing energy-critical nonlinear Schrödinger equation in $\mathbb{R}^{1+4}$, Amer. J. Math., 129 (2007), 1-60.  doi: 10.1353/ajm.2007.0004.

[38]

W. Schlag, Dispersive estimates for Schrödinger operators: A survey, Ann. of Math. Stud., 163 (2007), 255-285. 

[39]

A. Soffer and M. I. Weinstein, Resonances, radiation damping and instability in Hamiltonian nonlinear wave equations, Invent. Math., 136 (1999), 9-74.  doi: 10.1007/s002220050303.

[40]

H. Spohn, Kinetic equations from Hamiltonian dynamics, Rev. Mod. Phys., 52 (1980), 569-615.  doi: 10.1103/RevModPhys.52.569.

[41]

W. Strauss, Nonlinear scattering theory. Scattering theory in mathematical physics, Proceedings of the NATO Advanced Study Institue, (Denver, 1973), 53-78. NATO Advanced Science Institues, Volume C9. Reidel, Dordrecht, 1974.

[42]

W. Strauss, Nonlinear scattering theory at low energy: Sequel, J. Funct. Anal., 43 (1981), 281-293.  doi: 10.1016/0022-1236(81)90019-7.

[43]

T. Tao, Nonlinear Dispersive Equations: Local and Global Analysis American Mathematical Society, 2006. doi: 10.1090/cbms/106.

[44]

T. TaoM. Visan and X. Zhang, Minimal-mass blowup solutions of the mass-critical NLS, Forum Mathematicum, 20 (2008), 881-919.  doi: 10.1515/FORUM.2008.042.

[45]

T. TaoM. Visan and X. Zhang, Global well-posedness and scattering for the defocusing mass-critical nonlinear Schrödinger equation for radial data in high dimensions, Duke Math J., 140 (2007), 165-202.  doi: 10.1215/S0012-7094-07-14015-8.

[46]

M. C. Vilela, Inhomogeneous Strichartz estimates for the Schrödinger equation, Trans. Amer. Math. Soc., 359 (2007), 2123-2136.  doi: 10.1090/S0002-9947-06-04099-2.

[47]

N. Visciglia, On the decay of solutions to a class of defocusing NLS, Math. Res. Lett., 16 (2009), 919-926.  doi: 10.4310/MRL.2009.v16.n5.a14.

[48]

M. Visan, The defocusing energy-critical nonlinear Schrödinger equation in higher dimemsions, Duke Math. J., 138 (2007), 281-374.  doi: 10.1215/S0012-7094-07-13825-0.

show all references

References:
[1]

P. Alsholm and G. Schmidt, Spectral and scattering theory for Schrödinger operators, Arch. Rational Mech. Anal, 40 (1971), 281-311.  doi: 10.1007/BF00252679.

[2]

P. AntonelliR. Carles and J. D. Silva, Scattering for nonlinear Schrödinger equation under partial harmonic confinement, Comm. Math. Phys., 334 (2015), 367-396.  doi: 10.1007/s00220-014-2166-y.

[3]

V. Banica and N. Visciglia, Scattering for non linear Schrödinger equation with a delta potential, J. Differential Equations, 260 (2016), 4410-4439.  doi: 10.1016/j.jde.2015.11.016.

[4]

M. Beceanu and M. Goldberg, Schrödinger dispersive estimates for a scaling-critical class of potentials, Comm. Math. Phys., 314 (2012), 471-481.  doi: 10.1007/s00220-012-1435-x.

[5]

J. Bourgain, Global wellposedness of defocusing critical nonlinear Schrödinger equation in the radial case, J. Amer. Math. Soc., 12 (1999), 145-171.  doi: 10.1090/S0894-0347-99-00283-0.

[6]

P. Chen, J. Magniez and E. M. Ouhabaz, Riesz transforms on non-compact manifolds, arXiv: 1411.0137.

[7]

J. CollianderM. Czubak and J. Lee, Interaction Morawetz estimate for the magnetic Schrödinger equation and applications, Adv. Differential Equations, 19 (2014), 805-832. 

[8]

J. CollianderM. KeelG. StaffilaniH. Takaoka and T. Tao, Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in $\mathbb{R}^3$, Ann. of Math., 167 (2008), 767-865.  doi: 10.4007/annals.2008.167.767.

[9]

S. CuccagnaV. Georgiev and N. Visciglia, Decay and scattering of small solutions of pure power NLS in $\mathbb{R}$ with p>3 and with a potential, Comm. Pure Appl. Math., 67 (2014), 957-981.  doi: 10.1002/cpa.21465.

[10]

P. D'anconaL. FanelliL. Vega and N. Visciglia, Endpoint Strichartz estimates for the magnetic Schrödinger equation, J. Funct. Anal., 258 (2010), 3227-3240.  doi: 10.1016/j.jfa.2010.02.007.

[11]

P. D'ancona and V. Pierfelice, On the wave equation with a large rough potential, J. Funct. Anal., 227 (2005), 30-77.  doi: 10.1016/j.jfa.2005.05.013.

[12]

B. Dodson, Global well-posedness and scattering for the defocusing, L2-critical, nonlinear Schrödinger equation when d ≥ 3, J. Amer. Math. Soc., 25 (2012), 429-463.  doi: 10.1090/S0894-0347-2011-00727-3.

[13]

B. Dodson, Global well-posedness and scattering for the defocusing, L2-critical, nonlinear Schrödinger equation when d = 2, Duke Math. J., 165 (2016), no. 18,3435-3516.  doi: 10.1090/S0894-0347-2011-00727-3.

[14]

B. Dodson, Global well-posedness and scattering for the defocusing, L2-critical, nonlinear Schrödinger equation when d = 1 to appear in Amer. J. Math. , arXiv: 1010.0040. doi: 10.1090/S0894-0347-2011-00727-3.

[15]

B. Dodson, Global well-posedness and scattering for the mass critical nonlinear Schrödinger equation with mass below the mass of the ground state, Advances in Mathematics, 285 (2015), 1589-1618.  doi: 10.1016/j.aim.2015.04.030.

[16]

B. Dodson, Global well-posedness and scattering for the focusing, energy-critical nonlinear Schrödinger problem in dimension d = 4 for initial data below a ground state threshold, arXiv: 1409. 1950.

[17]

J. GinibreT. Ozawa and G. Velo, On the existence of the wave operators for a class of nonlinear Schrödinger equations, Ann. Inst. H. Poincare Phys. Theor., 60 (1994), 211-239. 

[18]

J. Ginibre and G. Velo, Scattering theory in the energy space for a class of nonlinear Schrödinger equations, J. Math. Pures Appl.(9), 64 (1985), 363-401. 

[19]

R. H. GoodmanR. E. Slusher and M. I. Weinstein, Stopping light on a defect, J. Opt. Soc. Am. B, 19 (2002), 1635-1652.  doi: 10.1364/JOSAB.19.001635.

[20]

R. H. GoodmanM. I. Weinstein and P. J. Holmes, Nonlinear propagation of light in one-dimensional periodic structures, J. Nonlinear Sci., 11 (2001), 123-168.  doi: 10.1007/s00332-001-0002-y.

[21]

Z. Hani and L. Thomann, Asymptotic behavior of the nonlinear Schrödinger equation with harmonic trapping, Comm. Pure Appl. Math., 69 (2016), 1727-1776.  doi: 10.1002/cpa.21594.

[22]

K. Hepp, The classical limit for quantum mechanical correlation functions, Comm. Math. Phys., 35 (1974), 265-277.  doi: 10.1007/BF01646348.

[23]

Y. Hong, Scattering for a nonlinear Schrödinger equation with a potential, Comm. Pure Appl. Anal.(5), 15 (2016), 1571-1601.  doi: 10.3934/cpaa.2016003.

[24]

S. IbrahimN. Masmoudi and K. Nakanishi, Scattering threshold for the focusing nonlinear Klein-Gordon equation, Anal. PDE, 4 (2011), 405-460.  doi: 10.2140/apde.2011.4.405.

[25]

J. L. JourneA. Soffer and C. D. Sogge, Decay estimates for Schrödinger operators, Comm. Pure Appl. Math., 44 (1991), 573-604.  doi: 10.1002/cpa.3160440504.

[26]

M. Keel and T. Tao, Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955-980.  doi: 10.1353/ajm.1998.0039.

[27]

S. Keraani, On the defect of compactness for the Strichartz estimates for the Schrödinger equations, J. Differential Equations, 175 (2001), 353-392.  doi: 10.1006/jdeq.2000.3951.

[28]

C. E. Kenig and F. Merle, Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case, Invent. Math., 166 (2006), 645-675.  doi: 10.1007/s00222-006-0011-4.

[29]

R. Killip and M. Visan, The focusing energy-critical nonlinear Schrödinger equation in dimensions five and higher, Amer. J. Math., 132 (2010), 361-424.  doi: 10.1353/ajm.0.0107.

[30]

R. KillipM. Visan and X. Zhang, The mass-critical nonlinear Schrödinger equation with radial data in dimensions three and higher, Analysis and PDE, 1 (2009), 229-266.  doi: 10.2140/apde.2008.1.229.

[31]

D. Lafontaine, Scattering for NLS with a potential on the line, Asymptotic Analysis, 100 (2016), 21-39.  doi: 10.3233/ASY-161384.

[32]

E. H. LiebR. Seiringer and J. Yngvason, A rigorous derivation of the Gross-Pitaevskii energy functional for a two-dimensional Bose gas, Comm. Math. Phys., 224 (2001), 17-31.  doi: 10.1007/s002200100533.

[33]

H. P. McKean and J. Shatah, The nonlinear Schrödinger equation and the nonlinear heat equation reduction to linear form, Comm. Pure Appl. Math., 44 (1991), 1067-1080.  doi: 10.1002/cpa.3160440817.

[34]

K. Nakanishi, Energy scattering for nonlinear Klein-Gordon and Schrödinger equations in spatial dimensions 1 and 2, Journal of Functional Analysis, 169 (1999), 201-225.  doi: 10.1006/jfan.1999.3503.

[35]

F. Planchon and L. Vega, Bilinear virial identities and applications, Ann. Sci. Ec. Norm. Super., 42 (2009), 261-290. 

[36]

I. Rodnianski and W. Schlag, Time decay for solutions of Schrödinger equations with rough and time-dependent potentials, Invent. Math., 155 (2004), 451-513.  doi: 10.1007/s00222-003-0325-4.

[37]

E. Ryckman and M. Visan, Global well-posedness and scattering for the defocusing energy-critical nonlinear Schrödinger equation in $\mathbb{R}^{1+4}$, Amer. J. Math., 129 (2007), 1-60.  doi: 10.1353/ajm.2007.0004.

[38]

W. Schlag, Dispersive estimates for Schrödinger operators: A survey, Ann. of Math. Stud., 163 (2007), 255-285. 

[39]

A. Soffer and M. I. Weinstein, Resonances, radiation damping and instability in Hamiltonian nonlinear wave equations, Invent. Math., 136 (1999), 9-74.  doi: 10.1007/s002220050303.

[40]

H. Spohn, Kinetic equations from Hamiltonian dynamics, Rev. Mod. Phys., 52 (1980), 569-615.  doi: 10.1103/RevModPhys.52.569.

[41]

W. Strauss, Nonlinear scattering theory. Scattering theory in mathematical physics, Proceedings of the NATO Advanced Study Institue, (Denver, 1973), 53-78. NATO Advanced Science Institues, Volume C9. Reidel, Dordrecht, 1974.

[42]

W. Strauss, Nonlinear scattering theory at low energy: Sequel, J. Funct. Anal., 43 (1981), 281-293.  doi: 10.1016/0022-1236(81)90019-7.

[43]

T. Tao, Nonlinear Dispersive Equations: Local and Global Analysis American Mathematical Society, 2006. doi: 10.1090/cbms/106.

[44]

T. TaoM. Visan and X. Zhang, Minimal-mass blowup solutions of the mass-critical NLS, Forum Mathematicum, 20 (2008), 881-919.  doi: 10.1515/FORUM.2008.042.

[45]

T. TaoM. Visan and X. Zhang, Global well-posedness and scattering for the defocusing mass-critical nonlinear Schrödinger equation for radial data in high dimensions, Duke Math J., 140 (2007), 165-202.  doi: 10.1215/S0012-7094-07-14015-8.

[46]

M. C. Vilela, Inhomogeneous Strichartz estimates for the Schrödinger equation, Trans. Amer. Math. Soc., 359 (2007), 2123-2136.  doi: 10.1090/S0002-9947-06-04099-2.

[47]

N. Visciglia, On the decay of solutions to a class of defocusing NLS, Math. Res. Lett., 16 (2009), 919-926.  doi: 10.4310/MRL.2009.v16.n5.a14.

[48]

M. Visan, The defocusing energy-critical nonlinear Schrödinger equation in higher dimemsions, Duke Math. J., 138 (2007), 281-374.  doi: 10.1215/S0012-7094-07-13825-0.

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