2017, 37(7): 3831-3866. doi: 10.3934/dcds.2017162

The energy-critical NLS with inverse-square potential

1. 

Department of Mathematics, UCLA, Los Angeles, CA 90095-1555, USA

2. 

Institute of Applied Physics and Computational Mathematics, Beijing 100088, China

3. 

Department of Mathematics, Beijing Institute of Technology, Beijing 100081, China

4. 

Laboratoire J. A. Dieudonné, Université Nice Sophia-Antipolis, 06108 Nice Cedex 02, France

Received  June 2016 Revised  March 2017 Published  April 2017

We consider the defocusing energy-critical nonlinear Schrödinger equation with inverse-square potential $iu_t = -Δ u + a|x|^{-2}u + |u|^4u$ in three space dimensions. We prove global well-posedness and scattering for $a > - \frac{1}{4} + \frac{1}{{25}}$. We also carry out the variational analysis needed to treat the focusing case.

Citation: Rowan Killip, Changxing Miao, Monica Visan, Junyong Zhang, Jiqiang Zheng. The energy-critical NLS with inverse-square potential. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 3831-3866. doi: 10.3934/dcds.2017162
References:
[1]

G. Andrews, R. Askey and R. Roy, Special Functions, Encyclopedia of Mathematics and its Applications, 71. Cambridge University Press, Cambridge, 1999. doi: 10.1017/CBO9781107325937.

[2]

T. Aubin, Problémes isopérimétriques et espaces de Sobolev, J. Diff. Geom., 11 (1976), 573-598. doi: 10.4310/jdg/1214433725.

[3]

H. Bahouri, P. Gérard, High frequency approximation of solutions to critical nonlinear wave equations, Amer. J. Math., 121 (1999), 131-175. doi: 10.1353/ajm.1999.0001.

[4]

G. A. Bliss, An integral inequality, J. London Math. Soc., 5 (1930), 40-46. doi: 10.1112/jlms/s1-5.1.40.

[5]

J. Bourgain, Global well-posedness of defocusing 3D critical NLS in the radial case, J. Amer. Math. Soc., 12 (1999), 145-171. doi: 10.1090/S0894-0347-99-00283-0.

[6]

H. Brézis, 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.

[7]

N. Burq, F. Planchon, J. Stalker, A. S. Tahvildar-Zadeh, Strichartz estimates for the wave and Schrödinger equations with the inverse-square potential, J. Funct. Anal., 203 (2003), 519-549. doi: 10.1016/S0022-1236(03)00238-6.

[8]

N. Burq, F. Planchon, J. Stalker, A. S. Tahvildar-Zadeh, Strichartz estimates for the wave and Schrödinger equations with potentials of critical decay, Indiana Univ. Math. J., 53 (2004), 1665-1680. doi: 10.1512/iumj.2004.53.2541.

[9]

M. Christ, A. Kiselev, Maximal functions associated to filtrations, J. Funct. Anal., 179 (2001), 409-425. doi: 10.1006/jfan.2000.3687.

[10]

M. Christ, M. Weinstein, Dispersion of small amplitude solutions of the generalized Korteweg-de Vries equation, J. Funct. Anal., 100 (1991), 87-109. doi: 10.1016/0022-1236(91)90103-C.

[11]

J. Colliander, M. Keel, G. Staffilani, H. Takaoka, T. Tao, Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in $\delta_1=\delta_1(d, \delta_0)$, Ann. of Math., 167 (2008), 767-865. doi: 10.4007/annals.2008.167.767.

[12]

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, preprint, arXiv: 1409.1950.

[13]

L. Fanelli, V. Felli, M. A. Fontelos, A. Primo, Time decay of scaling critical electromagnetic Schrödinger flows, Comm. Math. Phys., 324 (2013), 1033-1067. doi: 10.1007/s00220-013-1830-y.

[14]

P. Gérard, Description du défaut de compacité de l'injection de Sobolev, ESAIM Control Optim. Calc. Var., 3 (1998), 213-233. doi: 10.1051/cocv:1998107.

[15]

A. D. Ionescu, 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.

[16]

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

[17]

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

[18]

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

[19]

C. Jao, Energy-critical NLS with potentials of quadratic growth, preprint, arXiv: 1411.4950

[20]

H. Kalf, U. W. Schmincke, J. Walter, R. Wüst, On the spectral theory of Schrödinger and Dirac operators with strongly singular potentials, in Spectral theory and differential equations, Lect, Notes in Math., 448 (1975), 182-226.

[21]

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

[22]

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

[23]

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.

[24]

R. Killip, S. Kwon, S. Shao, M. Visan, On the mass-critical generalized KdV equation, Discrete Continuous Dynam. Systems -A, 32 (2012), 191-221. doi: 10.3934/dcds.2012.32.191.

[25]

R. Killip, C. Miao, M. Visan, J. Zhang and J. Zheng, Multipliers and Riesz transforms for the Schrödinger operator with inverse-square potential, preprint, arXiv: 1503.02716.

[26]

R. Killip, T. Oh, O. Pocovnicu and M. Visan, Solitons and scattering for the cubic-quintic nonlinear Schrödinger equation on ${\mathbb{R}}^3$ To appear in Arch. Ration. Mech. Anal. preprint, arXiv: 1409.6734.

[27]

R. Killip, B. Stovall, 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.

[28]

R. Killip, 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.

[29]

R. Killip, M. Visan, Nonlinear Schrödinger equations at critical regularity, in Evolution equations, Clay Math. Proc, Amer. Math. Soc., 17 (2013), 325-437.

[30]

R. Killip, M. Visan, Global well-posedness and scattering for the defocusing quintic NLS in three dimensions, Anal. PDE, 5 (2012), 855-885. doi: 10.2140/apde.2012.5.855.

[31]

R. Killip, M. Visan, 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.

[32]

H. Koch, D. Tataru and M. Visan, Dispersive Equations and Nonlinear Waves Oberwolfach Seminars, 45 Birkhauser/Springer Basel AG, Basel, 2014. doi: 10.1007/978-3-0348-0736-4.

[33]

E. Lieb and M. Loss, Analysis Second edition. Graduate Studies in Mathematics, 14 American Mathematical Society, Providence, RI, 2001. doi: 10.1090/gsm/014.

[34]

V. Liskevich, Z. Sobol, Estimates of integral kernels for semigroups associated with second order elliptic operators with singular coefficients, Potential Anal., 18 (2003), 359-390. doi: 10.1023/A:1021877025938.

[35]

P. D. Milman, Yu. A. Semenov, Global heat kernel bounds via desingularizing weights, J. Funct. Anal., 212 (2004), 373-398. doi: 10.1016/j.jfa.2003.12.008.

[36]

B. Pausader, N. Tzvetkov, X. Wang, Global regularity for the energy-critical NLS on $\mathbb{S}^3$, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 315-338. doi: 10.1016/j.anihpc.2013.03.006.

[37]

F. Planchon, J. Stalker, A. S. Tahvildar-Zadeh, Dispersive estimates for wave equation with the inverse-square potential, Discrete Contin. Dynam. Systems, 9 (2003), 1387-1400. doi: 10.3934/dcds.2003.9.1387.

[38]

M. Reed and B. Simon, Methods of Modern Mathematical Physics. Ⅱ. Fourier Analysis, Self-adjointness, Academic Press, New York-London, 1975.

[39]

E. Ryckman, 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.

[40]

G. Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura. Appl., 110 (1976), 353-372. doi: 10.1007/BF02418013.

[41]

T. Tao, Global well-posedness and scattering for higher-dimensional energy-critical non-linear Schrödinger equation for radial data, New York J. of Math., 11 (2005), 57-80.

[42]

T. Tao, M. Visan, Stability of energy-critical nonlinear Schrödinger equations in high dimensions, Electron. J. Diff. Eqns., 118 (2005), 1-28.

[43]

J. L. Vazquez, E. Zuazua, The Hardy inequality and the asymptotic behaviour of the heat equation with an inverse-square potential, J. Funct. Anal., 173 (2000), 103-153. doi: 10.1006/jfan.1999.3556.

[44]

M. Visan, The Defocusing Energy-Critical Nonlinear Schrödinger Equation in Dimensions Five and Higher Ph. D Thesis, UCLA, 2006.

[45]

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.

[46]

J. Zhang, J. Zheng, Scattering theory for nonlinear Schrödinger with inverse-square potential, J. Funct. Anal., 267 (2014), 2907-2932. doi: 10.1016/j.jfa.2014.08.012.

show all references

References:
[1]

G. Andrews, R. Askey and R. Roy, Special Functions, Encyclopedia of Mathematics and its Applications, 71. Cambridge University Press, Cambridge, 1999. doi: 10.1017/CBO9781107325937.

[2]

T. Aubin, Problémes isopérimétriques et espaces de Sobolev, J. Diff. Geom., 11 (1976), 573-598. doi: 10.4310/jdg/1214433725.

[3]

H. Bahouri, P. Gérard, High frequency approximation of solutions to critical nonlinear wave equations, Amer. J. Math., 121 (1999), 131-175. doi: 10.1353/ajm.1999.0001.

[4]

G. A. Bliss, An integral inequality, J. London Math. Soc., 5 (1930), 40-46. doi: 10.1112/jlms/s1-5.1.40.

[5]

J. Bourgain, Global well-posedness of defocusing 3D critical NLS in the radial case, J. Amer. Math. Soc., 12 (1999), 145-171. doi: 10.1090/S0894-0347-99-00283-0.

[6]

H. Brézis, 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.

[7]

N. Burq, F. Planchon, J. Stalker, A. S. Tahvildar-Zadeh, Strichartz estimates for the wave and Schrödinger equations with the inverse-square potential, J. Funct. Anal., 203 (2003), 519-549. doi: 10.1016/S0022-1236(03)00238-6.

[8]

N. Burq, F. Planchon, J. Stalker, A. S. Tahvildar-Zadeh, Strichartz estimates for the wave and Schrödinger equations with potentials of critical decay, Indiana Univ. Math. J., 53 (2004), 1665-1680. doi: 10.1512/iumj.2004.53.2541.

[9]

M. Christ, A. Kiselev, Maximal functions associated to filtrations, J. Funct. Anal., 179 (2001), 409-425. doi: 10.1006/jfan.2000.3687.

[10]

M. Christ, M. Weinstein, Dispersion of small amplitude solutions of the generalized Korteweg-de Vries equation, J. Funct. Anal., 100 (1991), 87-109. doi: 10.1016/0022-1236(91)90103-C.

[11]

J. Colliander, M. Keel, G. Staffilani, H. Takaoka, T. Tao, Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in $\delta_1=\delta_1(d, \delta_0)$, Ann. of Math., 167 (2008), 767-865. doi: 10.4007/annals.2008.167.767.

[12]

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, preprint, arXiv: 1409.1950.

[13]

L. Fanelli, V. Felli, M. A. Fontelos, A. Primo, Time decay of scaling critical electromagnetic Schrödinger flows, Comm. Math. Phys., 324 (2013), 1033-1067. doi: 10.1007/s00220-013-1830-y.

[14]

P. Gérard, Description du défaut de compacité de l'injection de Sobolev, ESAIM Control Optim. Calc. Var., 3 (1998), 213-233. doi: 10.1051/cocv:1998107.

[15]

A. D. Ionescu, 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.

[16]

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

[17]

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

[18]

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

[19]

C. Jao, Energy-critical NLS with potentials of quadratic growth, preprint, arXiv: 1411.4950

[20]

H. Kalf, U. W. Schmincke, J. Walter, R. Wüst, On the spectral theory of Schrödinger and Dirac operators with strongly singular potentials, in Spectral theory and differential equations, Lect, Notes in Math., 448 (1975), 182-226.

[21]

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

[22]

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

[23]

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.

[24]

R. Killip, S. Kwon, S. Shao, M. Visan, On the mass-critical generalized KdV equation, Discrete Continuous Dynam. Systems -A, 32 (2012), 191-221. doi: 10.3934/dcds.2012.32.191.

[25]

R. Killip, C. Miao, M. Visan, J. Zhang and J. Zheng, Multipliers and Riesz transforms for the Schrödinger operator with inverse-square potential, preprint, arXiv: 1503.02716.

[26]

R. Killip, T. Oh, O. Pocovnicu and M. Visan, Solitons and scattering for the cubic-quintic nonlinear Schrödinger equation on ${\mathbb{R}}^3$ To appear in Arch. Ration. Mech. Anal. preprint, arXiv: 1409.6734.

[27]

R. Killip, B. Stovall, 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.

[28]

R. Killip, 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.

[29]

R. Killip, M. Visan, Nonlinear Schrödinger equations at critical regularity, in Evolution equations, Clay Math. Proc, Amer. Math. Soc., 17 (2013), 325-437.

[30]

R. Killip, M. Visan, Global well-posedness and scattering for the defocusing quintic NLS in three dimensions, Anal. PDE, 5 (2012), 855-885. doi: 10.2140/apde.2012.5.855.

[31]

R. Killip, M. Visan, 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.

[32]

H. Koch, D. Tataru and M. Visan, Dispersive Equations and Nonlinear Waves Oberwolfach Seminars, 45 Birkhauser/Springer Basel AG, Basel, 2014. doi: 10.1007/978-3-0348-0736-4.

[33]

E. Lieb and M. Loss, Analysis Second edition. Graduate Studies in Mathematics, 14 American Mathematical Society, Providence, RI, 2001. doi: 10.1090/gsm/014.

[34]

V. Liskevich, Z. Sobol, Estimates of integral kernels for semigroups associated with second order elliptic operators with singular coefficients, Potential Anal., 18 (2003), 359-390. doi: 10.1023/A:1021877025938.

[35]

P. D. Milman, Yu. A. Semenov, Global heat kernel bounds via desingularizing weights, J. Funct. Anal., 212 (2004), 373-398. doi: 10.1016/j.jfa.2003.12.008.

[36]

B. Pausader, N. Tzvetkov, X. Wang, Global regularity for the energy-critical NLS on $\mathbb{S}^3$, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 315-338. doi: 10.1016/j.anihpc.2013.03.006.

[37]

F. Planchon, J. Stalker, A. S. Tahvildar-Zadeh, Dispersive estimates for wave equation with the inverse-square potential, Discrete Contin. Dynam. Systems, 9 (2003), 1387-1400. doi: 10.3934/dcds.2003.9.1387.

[38]

M. Reed and B. Simon, Methods of Modern Mathematical Physics. Ⅱ. Fourier Analysis, Self-adjointness, Academic Press, New York-London, 1975.

[39]

E. Ryckman, 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.

[40]

G. Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura. Appl., 110 (1976), 353-372. doi: 10.1007/BF02418013.

[41]

T. Tao, Global well-posedness and scattering for higher-dimensional energy-critical non-linear Schrödinger equation for radial data, New York J. of Math., 11 (2005), 57-80.

[42]

T. Tao, M. Visan, Stability of energy-critical nonlinear Schrödinger equations in high dimensions, Electron. J. Diff. Eqns., 118 (2005), 1-28.

[43]

J. L. Vazquez, E. Zuazua, The Hardy inequality and the asymptotic behaviour of the heat equation with an inverse-square potential, J. Funct. Anal., 173 (2000), 103-153. doi: 10.1006/jfan.1999.3556.

[44]

M. Visan, The Defocusing Energy-Critical Nonlinear Schrödinger Equation in Dimensions Five and Higher Ph. D Thesis, UCLA, 2006.

[45]

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.

[46]

J. Zhang, J. Zheng, Scattering theory for nonlinear Schrödinger with inverse-square potential, J. Funct. Anal., 267 (2014), 2907-2932. doi: 10.1016/j.jfa.2014.08.012.

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