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February  2018, 38(2): 563-587. doi: 10.3934/dcds.2018025

Energy-critical NLS with potentials of quadratic growth

Casey Jao

Department of Mathematics, UC Berkeley, Berkeley, CA 94720, USA

Received  April 2017 Published  February 2018

Fund Project: The first author is supported by NSF grants DMS-0838680, DMS-1265868, DMS-0901166, DMS-1161396.

We consider the global wellposedness problem for the nonlinear Schrödinger equation
$i\partial_t u = [-\tfrac{1}{2} Δ + V(x)] u ± |u|^{4/(d-2)} u, \ u(0)∈ Σ(\mathbf{R}^d),$
where
$Σ$
 is the weighted Sobolev space
$\dot{H}^1 \cap |x|^{-1} L^2$
. The case
$V(x) = \tfrac{1}{2}|x|^2$
 was recently treated by the author. This note generalizes the results to a class of "approximately quadratic" potentials.
We closely follow the previous concentration compactness arguments for the harmonic oscillator. A key technical difference is that in the absence of a concrete formula for the linear propagator, we apply more general tools from microlocal analysis, including a Fourier integral parametrix of Fujiwara.
Keywords: Energy-critical, profile decompositions, Schrödinger operators.
Mathematics Subject Classification: Primary: 35Q55; Secondary: 53C35.
Citation: Casey Jao. Energy-critical NLS with potentials of quadratic growth. Discrete & Continuous Dynamical Systems, 2018, 38 (2) : 563-587. doi: 10.3934/dcds.2018025
References:
[1]

K. Asada and D. Fujiwara, On some oscillatory integral transformations in $L^{2}(\textbf{R}^{n})$, Japan. J. Math. (N.S.), 4 (1978), 299-361.   Google Scholar

[2]

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

[3]

H. Brézis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486-490.  doi: 10.2307/2044999.  Google Scholar

[4]

R. Carles, Nonlinear schrödinger equation with time-dependent potential, Commun. Math Sci., 9 (2011), 937-964.  doi: 10.4310/CMS.2011.v9.n4.a1.  Google Scholar

[5]

J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in $\Bbb R^3$, Ann. of Math. (2), 167 (2008), 767-865.  doi: 10.4007/annals.2008.167.767.  Google Scholar

[6]

G. B. Folland, Harmonic Analysis in Phase Space vol. 122 of Annals of Mathematics Studies, Princeton University Press, Princeton, NJ, 1989. doi: 10.1515/9781400882427.  Google Scholar

[7]

D. Fujiwara, On the boundedness of integral transformations with highly oscillatory kernels, Proc. Japan Acad., 51 (1975), 96-99.  doi: 10.3792/pja/1195518693.  Google Scholar

[8]

D. Fujiwara, A construction of the fundamental solution for the Schrödinger equation, J. Analyse Math., 35 (1979), 41-96.  doi: 10.1007/BF02791062.  Google Scholar

[9]

D. Fujiwara, Remarks on convergence of the Feynman path integrals, Duke Math. J., 47 (1980), 559-600.  doi: 10.1215/S0012-7094-80-04734-1.  Google Scholar

[10]

W. Hebisch, A multiplier theorem for Schrödinger operators, Colloq. Math., 60/61 (1990), 659-664.  doi: 10.4064/cm-60-61-2-659-664.  Google Scholar

[11]

A. D. Ionescu and B. Pausader, The energy-critical defocusing NLS on $\mathbb{T}^3$, Duke Math. J., 161 (2012), 1581-1612.  doi: 10.1215/00127094-1593335.  Google Scholar

[12]

A. D. Ionescu and B. Pausader, Global well-posedness of the energy-critical defocusing {NLS} on $\mathbb{R}× \mathbb{T}^3$, Comm. Math. Phys., 312 (2012), 781-831.  doi: 10.1007/s00220-012-1474-3.  Google Scholar

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A. D. Ionescu, B. Pausader and G. Staffilani, On the global well-posedness of energy-critical {S}chrödinger equations in curved spaces, Anal. PDE, 5 (2012), 705-746.  doi: 10.2140/apde.2012.5.705.  Google Scholar

[14]

C. Jao, The energy-critical quantum harmonic oscillator, Comm. Partial Differential Equations, 41 (2016), 79-133.  doi: 10.1080/03605302.2015.1095767.  Google Scholar

[15]

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

[16]

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

[17]

S. Keraani, On the blow up phenomenon of the critical nonlinear Schrödinger equation, J. Funct. Anal., 235 (2006), 171-192.  doi: 10.1016/j.jfa.2005.10.005.  Google Scholar

[18]

R. Killip, S. Kwon, S. Shao and M. Visan, On the mass-critical generalized KdV equation, Discrete Contin. Dyn. Syst., 32 (2012), 191-221.  doi: 10.3934/dcds.2012.32.191.  Google Scholar

[19]

R. Killip, B. Stovall and M. Visan, Scattering for the cubic Klein-Gordon equation in two space dimensions, Trans. Amer. Math. Soc., 364 (2012), 1571-1631.  doi: 10.1090/S0002-9947-2011-05536-4.  Google Scholar

[20]

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

[21]

R. Killip and M. Vişan, Nonlinear Schrödinger equations at critical regularity, in Evolution equations, vol. 17 of Clay Math. Proc., Amer. Math. Soc., Providence, RI, 2013,325–437. Google Scholar

[22]

R. Killip, M. Visan and X. Zhang, Quintic NLS in the exterior of a strictly convex obstacle, Amer. J. Math., 138 (2016), 1193-1346.  doi: 10.1353/ajm.2016.0039.  Google Scholar

[23]

R. Killip, M. Visan and X. Zhang, Energy-critical NLS with quadratic potentials, Comm. Partial Differential Equations, 34 (2009), 1531-1565.  doi: 10.1080/03605300903328109.  Google Scholar

[24]

Y.-G. Oh, Cauchy problem and Ehrenfest's law of nonlinear Schrödinger equations with potentials, J. Differential Equations, 81 (1989), 255-274.  doi: 10.1016/0022-0396(89)90123-X.  Google Scholar

[25]

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

[26]

M. E. Taylor, Tools for PDE vol. 81 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 2000, Pseudodifferential operators, paradifferential operators, and layer potentials.  Google Scholar

[27]

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

[28]

J. Zhang, Stability of attractive Bose-Einstein condensates, J. Statist. Phys., 101 (2000), 731-746.  doi: 10.1023/A:1026437923987.  Google Scholar

show all references

References:
[1]

K. Asada and D. Fujiwara, On some oscillatory integral transformations in $L^{2}(\textbf{R}^{n})$, Japan. J. Math. (N.S.), 4 (1978), 299-361.   Google Scholar

[2]

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

[3]

H. Brézis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486-490.  doi: 10.2307/2044999.  Google Scholar

[4]

R. Carles, Nonlinear schrödinger equation with time-dependent potential, Commun. Math Sci., 9 (2011), 937-964.  doi: 10.4310/CMS.2011.v9.n4.a1.  Google Scholar

[5]

J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in $\Bbb R^3$, Ann. of Math. (2), 167 (2008), 767-865.  doi: 10.4007/annals.2008.167.767.  Google Scholar

[6]

G. B. Folland, Harmonic Analysis in Phase Space vol. 122 of Annals of Mathematics Studies, Princeton University Press, Princeton, NJ, 1989. doi: 10.1515/9781400882427.  Google Scholar

[7]

D. Fujiwara, On the boundedness of integral transformations with highly oscillatory kernels, Proc. Japan Acad., 51 (1975), 96-99.  doi: 10.3792/pja/1195518693.  Google Scholar

[8]

D. Fujiwara, A construction of the fundamental solution for the Schrödinger equation, J. Analyse Math., 35 (1979), 41-96.  doi: 10.1007/BF02791062.  Google Scholar

[9]

D. Fujiwara, Remarks on convergence of the Feynman path integrals, Duke Math. J., 47 (1980), 559-600.  doi: 10.1215/S0012-7094-80-04734-1.  Google Scholar

[10]

W. Hebisch, A multiplier theorem for Schrödinger operators, Colloq. Math., 60/61 (1990), 659-664.  doi: 10.4064/cm-60-61-2-659-664.  Google Scholar

[11]

A. D. Ionescu and B. Pausader, The energy-critical defocusing NLS on $\mathbb{T}^3$, Duke Math. J., 161 (2012), 1581-1612.  doi: 10.1215/00127094-1593335.  Google Scholar

[12]

A. D. Ionescu and B. Pausader, Global well-posedness of the energy-critical defocusing {NLS} on $\mathbb{R}× \mathbb{T}^3$, Comm. Math. Phys., 312 (2012), 781-831.  doi: 10.1007/s00220-012-1474-3.  Google Scholar

[13]

A. D. Ionescu, B. Pausader and G. Staffilani, On the global well-posedness of energy-critical {S}chrödinger equations in curved spaces, Anal. PDE, 5 (2012), 705-746.  doi: 10.2140/apde.2012.5.705.  Google Scholar

[14]

C. Jao, The energy-critical quantum harmonic oscillator, Comm. Partial Differential Equations, 41 (2016), 79-133.  doi: 10.1080/03605302.2015.1095767.  Google Scholar

[15]

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

[16]

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

[17]

S. Keraani, On the blow up phenomenon of the critical nonlinear Schrödinger equation, J. Funct. Anal., 235 (2006), 171-192.  doi: 10.1016/j.jfa.2005.10.005.  Google Scholar

[18]

R. Killip, S. Kwon, S. Shao and M. Visan, On the mass-critical generalized KdV equation, Discrete Contin. Dyn. Syst., 32 (2012), 191-221.  doi: 10.3934/dcds.2012.32.191.  Google Scholar

[19]

R. Killip, B. Stovall and M. Visan, Scattering for the cubic Klein-Gordon equation in two space dimensions, Trans. Amer. Math. Soc., 364 (2012), 1571-1631.  doi: 10.1090/S0002-9947-2011-05536-4.  Google Scholar

[20]

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

[21]

R. Killip and M. Vişan, Nonlinear Schrödinger equations at critical regularity, in Evolution equations, vol. 17 of Clay Math. Proc., Amer. Math. Soc., Providence, RI, 2013,325–437. Google Scholar

[22]

R. Killip, M. Visan and X. Zhang, Quintic NLS in the exterior of a strictly convex obstacle, Amer. J. Math., 138 (2016), 1193-1346.  doi: 10.1353/ajm.2016.0039.  Google Scholar

[23]

R. Killip, M. Visan and X. Zhang, Energy-critical NLS with quadratic potentials, Comm. Partial Differential Equations, 34 (2009), 1531-1565.  doi: 10.1080/03605300903328109.  Google Scholar

[24]

Y.-G. Oh, Cauchy problem and Ehrenfest's law of nonlinear Schrödinger equations with potentials, J. Differential Equations, 81 (1989), 255-274.  doi: 10.1016/0022-0396(89)90123-X.  Google Scholar

[25]

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

[26]

M. E. Taylor, Tools for PDE vol. 81 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 2000, Pseudodifferential operators, paradifferential operators, and layer potentials.  Google Scholar

[27]

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

[28]

J. Zhang, Stability of attractive Bose-Einstein condensates, J. Statist. Phys., 101 (2000), 731-746.  doi: 10.1023/A:1026437923987.  Google Scholar

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