Scattering of solutions to the nonlinear Schrödinger equations with regular potentials

Xing Cheng; Ze Li; Lifeng Zhao

doi:10.3934/dcds.2017129

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

Scattering of solutions to the nonlinear Schrödinger equations with regular potentials

Xing Cheng ^1, , Ze Li ^2, and Lifeng Zhao ^2,,

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.

Keywords: Nonlinear Schrödinger equations, potential, scattering, energy critical.

Mathematics Subject Classification: 35Q55.

Citation: Xing Cheng, Ze Li, Lifeng Zhao. Scattering of solutions to the nonlinear Schrödinger equations with regular potentials. Discrete & 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. Google Scholar
[2]	P. Antonelli, R. 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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[6]	P. Chen, J. Magniez and E. M. Ouhabaz, Riesz transforms on non-compact manifolds, arXiv: 1411.0137. Google Scholar
[7]	J. Colliander, M. Czubak and J. Lee, Interaction Morawetz estimate for the magnetic Schrödinger equation and applications, Adv. Differential Equations, 19 (2014), 805-832. Google Scholar
[8]	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., 167 (2008), 767-865. doi: 10.4007/annals.2008.167.767. Google Scholar
[9]	S. Cuccagna, V. 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. Google Scholar
[10]	P. D'ancona, L. Fanelli, L. 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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
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[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. Google Scholar
[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. Google Scholar
[17]	J. Ginibre, T. 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. Google Scholar
[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. Google Scholar
[19]	R. H. Goodman, R. 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. Google Scholar
[20]	R. H. Goodman, M. 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. Google Scholar
[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. Google Scholar
[22]	K. Hepp, The classical limit for quantum mechanical correlation functions, Comm. Math. Phys., 35 (1974), 265-277. doi: 10.1007/BF01646348. Google Scholar
[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. Google Scholar
[24]	S. Ibrahim, N. 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. Google Scholar
[25]	J. L. Journe, A. Soffer and C. D. Sogge, Decay estimates for Schrödinger operators, Comm. Pure Appl. Math., 44 (1991), 573-604. doi: 10.1002/cpa.3160440504. Google Scholar
[26]	M. Keel and T. Tao, Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955-980. doi: 10.1353/ajm.1998.0039. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[30]	R. Killip, M. 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. Google Scholar
[31]	D. Lafontaine, Scattering for NLS with a potential on the line, Asymptotic Analysis, 100 (2016), 21-39. doi: 10.3233/ASY-161384. Google Scholar
[32]	E. H. Lieb, R. 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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[35]	F. Planchon and L. Vega, Bilinear virial identities and applications, Ann. Sci. Ec. Norm. Super., 42 (2009), 261-290. Google Scholar
[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. Google Scholar
[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. Google Scholar
[38]	W. Schlag, Dispersive estimates for Schrödinger operators: A survey, Ann. of Math. Stud., 163 (2007), 255-285. Google Scholar
[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. Google Scholar
[40]	H. Spohn, Kinetic equations from Hamiltonian dynamics, Rev. Mod. Phys., 52 (1980), 569-615. doi: 10.1103/RevModPhys.52.569. Google Scholar
[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. Google Scholar
[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. Google Scholar
[43]	T. Tao, Nonlinear Dispersive Equations: Local and Global Analysis American Mathematical Society, 2006. doi: 10.1090/cbms/106. Google Scholar
[44]	T. Tao, M. 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. Google Scholar
[45]	T. Tao, M. 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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar

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. Google Scholar
[2]	P. Antonelli, R. 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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[6]	P. Chen, J. Magniez and E. M. Ouhabaz, Riesz transforms on non-compact manifolds, arXiv: 1411.0137. Google Scholar
[7]	J. Colliander, M. Czubak and J. Lee, Interaction Morawetz estimate for the magnetic Schrödinger equation and applications, Adv. Differential Equations, 19 (2014), 805-832. Google Scholar
[8]	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., 167 (2008), 767-865. doi: 10.4007/annals.2008.167.767. Google Scholar
[9]	S. Cuccagna, V. 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. Google Scholar
[10]	P. D'ancona, L. Fanelli, L. 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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[17]	J. Ginibre, T. 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. Google Scholar
[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. Google Scholar
[19]	R. H. Goodman, R. 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. Google Scholar
[20]	R. H. Goodman, M. 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. Google Scholar
[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. Google Scholar
[22]	K. Hepp, The classical limit for quantum mechanical correlation functions, Comm. Math. Phys., 35 (1974), 265-277. doi: 10.1007/BF01646348. Google Scholar
[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. Google Scholar
[24]	S. Ibrahim, N. 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. Google Scholar
[25]	J. L. Journe, A. Soffer and C. D. Sogge, Decay estimates for Schrödinger operators, Comm. Pure Appl. Math., 44 (1991), 573-604. doi: 10.1002/cpa.3160440504. Google Scholar
[26]	M. Keel and T. Tao, Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955-980. doi: 10.1353/ajm.1998.0039. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[30]	R. Killip, M. 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. Google Scholar
[31]	D. Lafontaine, Scattering for NLS with a potential on the line, Asymptotic Analysis, 100 (2016), 21-39. doi: 10.3233/ASY-161384. Google Scholar
[32]	E. H. Lieb, R. 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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[35]	F. Planchon and L. Vega, Bilinear virial identities and applications, Ann. Sci. Ec. Norm. Super., 42 (2009), 261-290. Google Scholar
[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. Google Scholar
[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. Google Scholar
[38]	W. Schlag, Dispersive estimates for Schrödinger operators: A survey, Ann. of Math. Stud., 163 (2007), 255-285. Google Scholar
[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. Google Scholar
[40]	H. Spohn, Kinetic equations from Hamiltonian dynamics, Rev. Mod. Phys., 52 (1980), 569-615. doi: 10.1103/RevModPhys.52.569. Google Scholar
[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. Google Scholar
[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. Google Scholar
[43]	T. Tao, Nonlinear Dispersive Equations: Local and Global Analysis American Mathematical Society, 2006. doi: 10.1090/cbms/106. Google Scholar
[44]	T. Tao, M. 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. Google Scholar
[45]	T. Tao, M. 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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar

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