July 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 and 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 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.

[7]

N. BurqF. PlanchonJ. Stalker and 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. BurqF. PlanchonJ. Stalker and 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 and A. Kiselev, Maximal functions associated to filtrations, J. Funct. Anal., 179 (2001), 409-425. doi: 10.1006/jfan.2000.3687.

[10]

M. Christ and 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. CollianderM. KeelG. StaffilaniH. Takaoka and 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. FanelliV. FelliM. A. Fontelos and 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 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.

[16]

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.

[17]

A. D. IonescuB. Pausader and 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. KalfU. W. SchminckeJ. Walter and 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 and T. Tao, Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955-980. doi: 10.1353/ajm.1998.0039.

[22]

C. Kenig and 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. KillipS. KwonS. Shao and 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. KillipB. 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.

[28]

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.

[29]

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

[30]

R. Killip and 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. KillipM. 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.

[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 and 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 and 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. PausaderN. Tzvetkov and 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. PlanchonJ. Stalker and 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 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.

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

[43]

J. L. Vazquez and 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 and 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 and 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 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.

[7]

N. BurqF. PlanchonJ. Stalker and 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. BurqF. PlanchonJ. Stalker and 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 and A. Kiselev, Maximal functions associated to filtrations, J. Funct. Anal., 179 (2001), 409-425. doi: 10.1006/jfan.2000.3687.

[10]

M. Christ and 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. CollianderM. KeelG. StaffilaniH. Takaoka and 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. FanelliV. FelliM. A. Fontelos and 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 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.

[16]

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.

[17]

A. D. IonescuB. Pausader and 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. KalfU. W. SchminckeJ. Walter and 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 and T. Tao, Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955-980. doi: 10.1353/ajm.1998.0039.

[22]

C. Kenig and 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. KillipS. KwonS. Shao and 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. KillipB. 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.

[28]

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.

[29]

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

[30]

R. Killip and 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. KillipM. 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.

[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 and 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 and 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. PausaderN. Tzvetkov and 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. PlanchonJ. Stalker and 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 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.

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

[43]

J. L. Vazquez and 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 and 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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