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. Google Scholar

[2]

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

[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. Google Scholar

[4]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[9]

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

[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. Google Scholar

[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. Google Scholar

[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.Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[18]

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

[19]

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

[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. Google Scholar

[21]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.Google Scholar

[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.Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[38]

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

[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. Google Scholar

[40]

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

[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. Google Scholar

[42]

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

[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. Google Scholar

[44]

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

[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. Google Scholar

[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. Google Scholar

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. Google Scholar

[2]

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

[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. Google Scholar

[4]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[9]

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

[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. Google Scholar

[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. Google Scholar

[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.Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[18]

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

[19]

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

[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. Google Scholar

[21]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.Google Scholar

[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.Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[38]

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

[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. Google Scholar

[40]

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

[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. Google Scholar

[42]

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

[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. Google Scholar

[44]

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

[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. Google Scholar

[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. Google Scholar

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