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November  2016, 15(6): 2023-2058. doi: 10.3934/cpaa.2016026

Global dynamics above the ground state for the energy-critical Schrödinger equation with radial data

Kenji Nakanishi 1, and  Tristan Roy 2,

1. 

Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology Osaka University, Suita, Osaka 565-0871, Japan
2. 

Graduate School of Mathematics, Nagoya University, Japan

Received  October 2015 Revised  June 2016 Published  September 2016

Consider the focusing energy critical Schrödinger equation in three space dimensions with radial initial data in the energy space. We describe the global dynamics of all the solutions of which the energy is at most slightly larger than that of the ground states, according to whether it stays in a neighborhood of them, blows up in finite time or scatters. In analogy with [19], the proof uses an analysis of the hyperbolic dynamics near them and the variational structure far from them. The key step that allows to classify the solutions is the one-pass lemma. The main difference between [19] and this paper is that one has to introduce a scaling parameter in order to describe the dynamics near them. One has to take into account this parameter in the analysis around the ground states by introducing some orthogonality conditions. One also has to take it into account in the proof of the one-pass lemma by comparing the contribution in the variational region and in the hyperbolic region.
Keywords: blow-up, solitons, Sobolev critical exponent., scattering theory, nonlinear Schrödinger equation.
Mathematics Subject Classification: Primary: 35Q5.
Citation: Kenji Nakanishi, Tristan Roy. Global dynamics above the ground state for the energy-critical Schrödinger equation with radial data. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2023-2058. doi: 10.3934/cpaa.2016026
References:
[1]

T. Aubin, Equations differentielles non lineaires et probleme de Yamabe concernant la courbure scalaire, J. Math. Pures. Appl., 55 (1976), 269-296.  Google Scholar

[2]

H. Bahouri and P. Gerard, High frequency approximation of solutions to critical nonlinear wave equations, Amer. J. Math., 121 (1999), 131-175.  Google Scholar

[3]

J. Bourgain, Global well-posedness of defocusing $3D$ critical NLS in the radial case, J. Amer. Math. Soc., 12 (1999), 145-171. Google Scholar

[4]

T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, 10, New York University, Courant Institute of Mathematical Sciences, New York, 2003. doi: 10.1090/cln/010.  Google Scholar

[5]

T. Cazenave and F. B. Weissler, The Cauchy problem for the critical nonlinear Schrödinger equation in $H^s$, Nonlinear Anal., 14 (1990), 807-836. doi: 10.1016/0362-546X(90)90023-A.  Google Scholar

[6]

T. Duyckaerts and F. Merle, Dynamic of threshold solutions for energy-critical NLS, Geom. Funct. Anal., 18 (2009), 1787-1840. doi: 10.1007/s00039-009-0707-x.  Google Scholar

[7]

P. Gerard, Y. Meyer and F. Oru, Inégalités de Sobolev précisées,, \emph{S\'eminaire E.D.P} (1996-1997), (): 1996.   Google Scholar

[8]

M. Grillakis, Analysis of the linearization around a critical point of an infinite-dimensional Hamiltonian system, Comm. Pure. Appl. Math., 43 (1990), 299-333. doi: 10.1002/cpa.3160430302.  Google Scholar

[9]

M. Keel and T. Tao, Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955-980.  Google Scholar

[10]

S. Keraani, On the defect of compactness for the Strichartz estimates of the Schrödinger equations, J. Diff. Eq., 175 (2001), 353-392. doi: 10.1006/jdeq.2000.3951.  Google Scholar

[11]

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

[12]

J. Krieger, K. Nakanishi and W. Schlag, Global dynamics away from the ground state for the energy-critical nonlinear wave equation, Amer. J. Math., 135 (2013), 935-965. doi: 10.1353/ajm.2013.0034.  Google Scholar

[13]

J. Krieger, K. Nakanishi and W. Schlag, Global dynamics of the nonradial energy-critical wave equation above the ground state energy, Discrete, Cont, Dyn. Syst., 33 (2013), 2423-2450.  Google Scholar

[14]

P. L. Lions, The concentration-compactness principle in the calculus of variations. (The limit case, Part I.), Rev. Mat. Iberoamericana, 1 (1985), 145-201. doi: 10.4171/RMI/6.  Google Scholar

[15]

F. Merle and L. Vega, Compactness at blow-up time for $L^{2}$ solutions of the critical nonlinear Schrödinger equation in $2D$, Internat. Math. Res. Notices, 1998, 399-425. doi: 10.1155/S1073792898000270.  Google Scholar

[16]

K. Nakanishi, Energy scattering for nonlinear Klein-Gordon equation and Schrödinger equation in spatial dimensions $1$ and $2$, J. Funct. Anal., 169 (1999), 201-225. doi: 10.1006/jfan.1999.3503.  Google Scholar

[17]

K. Nakanishi and W. Schlag, Global dynamics above the ground state energy for the focusing nonlinear Klein-Gordon equation, J. Diff. Eq., 250 (2011), 2299-2333. doi: 10.1016/j.jde.2010.10.027.  Google Scholar

[18]

K. Nakanishi and W. Schlag, Global dynamics above the ground state energy for the focusing nonlinear Klein-Gordon equation without a radial assumption, Arch. Rational Mech. Analysis, 203 (2012), 809-851. doi: 10.1007/s00205-011-0462-7.  Google Scholar

[19]

K. Nakanishi and W. Schlag, Global dynamics above the ground state energy for the cubic NLS equation in 3D, Calc. Var. and PDE, 44 (2012), 1-45. doi: 10.1007/s00526-011-0424-9.  Google Scholar

[20]

K. Nakanishi and W. Schlag, Invariant Manifolds and Dispersive Hamiltonian Evolution Equations, Zurich Lectures in Advanced Mathematics, EMS, 2011. doi: 10.4171/095.  Google Scholar

[21]

T. Ogawa and Y. Tsutsumi, Blow-up of $H^{1}$ solution for the nonlinear Schrödinger qquation, J. Diff. Eq., 92 (1991), 317-330. doi: 10.1016/0022-0396(91)90052-B.  Google Scholar

[22]

W. Schlag, Spectral theory and nonlinear differential equations: a survey, Discrete, Cont, Dyn. Syst., 15 (2006), 703-723. doi: 10.3934/dcds.2006.15.703.  Google Scholar

[23]

G. Talenti, Best Constant In Sobolev Inequality, Ann. Mat. Pura. Appl., 110 (1976), 353-372.  Google Scholar

show all references

References:
[1]

T. Aubin, Equations differentielles non lineaires et probleme de Yamabe concernant la courbure scalaire, J. Math. Pures. Appl., 55 (1976), 269-296.  Google Scholar

[2]

H. Bahouri and P. Gerard, High frequency approximation of solutions to critical nonlinear wave equations, Amer. J. Math., 121 (1999), 131-175.  Google Scholar

[3]

J. Bourgain, Global well-posedness of defocusing $3D$ critical NLS in the radial case, J. Amer. Math. Soc., 12 (1999), 145-171. Google Scholar

[4]

T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, 10, New York University, Courant Institute of Mathematical Sciences, New York, 2003. doi: 10.1090/cln/010.  Google Scholar

[5]

T. Cazenave and F. B. Weissler, The Cauchy problem for the critical nonlinear Schrödinger equation in $H^s$, Nonlinear Anal., 14 (1990), 807-836. doi: 10.1016/0362-546X(90)90023-A.  Google Scholar

[6]

T. Duyckaerts and F. Merle, Dynamic of threshold solutions for energy-critical NLS, Geom. Funct. Anal., 18 (2009), 1787-1840. doi: 10.1007/s00039-009-0707-x.  Google Scholar

[7]

P. Gerard, Y. Meyer and F. Oru, Inégalités de Sobolev précisées,, \emph{S\'eminaire E.D.P} (1996-1997), (): 1996.   Google Scholar

[8]

M. Grillakis, Analysis of the linearization around a critical point of an infinite-dimensional Hamiltonian system, Comm. Pure. Appl. Math., 43 (1990), 299-333. doi: 10.1002/cpa.3160430302.  Google Scholar

[9]

M. Keel and T. Tao, Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955-980.  Google Scholar

[10]

S. Keraani, On the defect of compactness for the Strichartz estimates of the Schrödinger equations, J. Diff. Eq., 175 (2001), 353-392. doi: 10.1006/jdeq.2000.3951.  Google Scholar

[11]

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

[12]

J. Krieger, K. Nakanishi and W. Schlag, Global dynamics away from the ground state for the energy-critical nonlinear wave equation, Amer. J. Math., 135 (2013), 935-965. doi: 10.1353/ajm.2013.0034.  Google Scholar

[13]

J. Krieger, K. Nakanishi and W. Schlag, Global dynamics of the nonradial energy-critical wave equation above the ground state energy, Discrete, Cont, Dyn. Syst., 33 (2013), 2423-2450.  Google Scholar

[14]

P. L. Lions, The concentration-compactness principle in the calculus of variations. (The limit case, Part I.), Rev. Mat. Iberoamericana, 1 (1985), 145-201. doi: 10.4171/RMI/6.  Google Scholar

[15]

F. Merle and L. Vega, Compactness at blow-up time for $L^{2}$ solutions of the critical nonlinear Schrödinger equation in $2D$, Internat. Math. Res. Notices, 1998, 399-425. doi: 10.1155/S1073792898000270.  Google Scholar

[16]

K. Nakanishi, Energy scattering for nonlinear Klein-Gordon equation and Schrödinger equation in spatial dimensions $1$ and $2$, J. Funct. Anal., 169 (1999), 201-225. doi: 10.1006/jfan.1999.3503.  Google Scholar

[17]

K. Nakanishi and W. Schlag, Global dynamics above the ground state energy for the focusing nonlinear Klein-Gordon equation, J. Diff. Eq., 250 (2011), 2299-2333. doi: 10.1016/j.jde.2010.10.027.  Google Scholar

[18]

K. Nakanishi and W. Schlag, Global dynamics above the ground state energy for the focusing nonlinear Klein-Gordon equation without a radial assumption, Arch. Rational Mech. Analysis, 203 (2012), 809-851. doi: 10.1007/s00205-011-0462-7.  Google Scholar

[19]

K. Nakanishi and W. Schlag, Global dynamics above the ground state energy for the cubic NLS equation in 3D, Calc. Var. and PDE, 44 (2012), 1-45. doi: 10.1007/s00526-011-0424-9.  Google Scholar

[20]

K. Nakanishi and W. Schlag, Invariant Manifolds and Dispersive Hamiltonian Evolution Equations, Zurich Lectures in Advanced Mathematics, EMS, 2011. doi: 10.4171/095.  Google Scholar

[21]

T. Ogawa and Y. Tsutsumi, Blow-up of $H^{1}$ solution for the nonlinear Schrödinger qquation, J. Diff. Eq., 92 (1991), 317-330. doi: 10.1016/0022-0396(91)90052-B.  Google Scholar

[22]

W. Schlag, Spectral theory and nonlinear differential equations: a survey, Discrete, Cont, Dyn. Syst., 15 (2006), 703-723. doi: 10.3934/dcds.2006.15.703.  Google Scholar

[23]

G. Talenti, Best Constant In Sobolev Inequality, Ann. Mat. Pura. Appl., 110 (1976), 353-372.  Google Scholar

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