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

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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

Abstract
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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 and 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.
[2]	H. Bahouri and P. Gerard, High frequency approximation of solutions to critical nonlinear wave equations, Amer. J. Math., 121 (1999), 131-175.
[3]	J. Bourgain, Global well-posedness of defocusing $3D$ critical NLS in the radial case, J. Amer. Math. Soc., 12 (1999), 145-171.
[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.
[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.
[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.
[7]	P. Gerard, Y. Meyer and F. Oru, Inégalités de Sobolev précisées,, \emph{S\'eminaire E.D.P} (1996-1997), (): 1996.
[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.
[9]	M. Keel and T. Tao, Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955-980.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[20]	K. Nakanishi and W. Schlag, Invariant Manifolds and Dispersive Hamiltonian Evolution Equations, Zurich Lectures in Advanced Mathematics, EMS, 2011. doi: 10.4171/095.
[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.
[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.
[23]	G. Talenti, Best Constant In Sobolev Inequality, Ann. Mat. Pura. Appl., 110 (1976), 353-372.

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References:

[1]	T. Aubin, Equations differentielles non lineaires et probleme de Yamabe concernant la courbure scalaire, J. Math. Pures. Appl., 55 (1976), 269-296.
[2]	H. Bahouri and P. Gerard, High frequency approximation of solutions to critical nonlinear wave equations, Amer. J. Math., 121 (1999), 131-175.
[3]	J. Bourgain, Global well-posedness of defocusing $3D$ critical NLS in the radial case, J. Amer. Math. Soc., 12 (1999), 145-171.
[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.
[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.
[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.
[7]	P. Gerard, Y. Meyer and F. Oru, Inégalités de Sobolev précisées,, \emph{S\'eminaire E.D.P} (1996-1997), (): 1996.
[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.
[9]	M. Keel and T. Tao, Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955-980.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[20]	K. Nakanishi and W. Schlag, Invariant Manifolds and Dispersive Hamiltonian Evolution Equations, Zurich Lectures in Advanced Mathematics, EMS, 2011. doi: 10.4171/095.
[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.
[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.
[23]	G. Talenti, Best Constant In Sobolev Inequality, Ann. Mat. Pura. Appl., 110 (1976), 353-372.

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