July  2015, 14(4): 1275-1326. doi: 10.3934/cpaa.2015.14.1275

Profiles for bounded solutions of dispersive equations, with applications to energy-critical wave and Schrödinger equations

1. 

LAGA (UMR 7539), Institut Galilée, Université Paris 13, 99, avenue Jean-Baptiste Clément, 93430 Villetaneuse, France

2. 

Department of Mathematics, University of Chicago, Chicago, Illinois, 60637–1514, United States

3. 

Université de Cergy-Pontoise and IHES, Laboratoire de mathématiques, UMR CNRS 8088, 2, av. Adolphe Chauvin, 95302 Cergy-Pontoise cedex

Received  October 2013 Revised  March 2014 Published  April 2015

Consider a bounded solution of the focusing, energy-critical wave equation that does not scatter to a linear solution. We prove that this solution converges in some weak sense, along a sequence of times and up to scaling and space translation, to a sum of solitary waves. This result is a consequence of a new general compactness/rigidity argument based on profile decomposition. We also give an application of this method to the energy-critical Schrödinger equation.
Citation: Thomas Duyckaerts, Carlos E. Kenig, Frank Merle. Profiles for bounded solutions of dispersive equations, with applications to energy-critical wave and Schrödinger equations. Communications on Pure & Applied Analysis, 2015, 14 (4) : 1275-1326. doi: 10.3934/cpaa.2015.14.1275
References:
[1]

Takafumi Akahori and Hayato Nawa, Blowup and scattering problems for the nonlinear Schrödinger equations,, \emph{Kyoto J. Math.}, 53 (2013), 629. doi: 10.1215/21562261-2265914.

[2]

Bulut Aynur, Maximizers for the Strichartz inequalities for the wave equation,, \emph{Differential Integral Equations}, 23 (2010), 1035.

[3]

Hajer Bahouri and Patrick Gérard, High frequency approximation of solutions to critical nonlinear wave equations,, \emph{Amer. J. Math.}, 121 (1999), 131.

[4]

Aynur Bulut, Magdalena Czubak, Dong Li, Nataša Pavlović and Xiaoyi Zhang, Stability and unconditional uniqueness of solutions for energy critical wave equations in high dimensions,, \emph{Comm. Partial Differential Equations}, 38 (2013), 575. doi: 10.1080/03605302.2012.756520.

[5]

Thierry Cazenave and Fred B. Weissler, The Cauchy problem for the critical nonlinear Schrödinger equation in $H^s$,, \emph{Nonlinear Anal.}, 14 (1990), 807. doi: 10.1016/0362-546X(90)90023-A.

[6]

Demetrios Christodoulou and A. Shadi Tahvildar-Zadeh, On the asymptotic behavior of spherically symmetric wave maps,, \emph{Duke Math. J.}, 71 (1993), 31. doi: 10.1215/S0012-7094-93-07103-7.

[7]

Peter Constantin and Jean-Claude Saut, Local smoothing properties of Schrödinger equations,, \emph{Indiana Univ. Math. J.}, 38 (1989), 791. doi: 10.1512/iumj.1989.38.38037.

[8]

Raphaël Côte, Soliton resolution for equivariant wave maps to the sphere,, Preprint, ().

[9]

Manuel del Pino, Personal communication., \quad, ().

[10]

Manuel del Pino, Monica Musso, Frank Pacard and Angela Pistoia, Large energy entire solutions for the Yamabe equation,, \emph{J. Differential Equations}, 251 (2011), 2568. doi: 10.1016/j.jde.2011.03.008.

[11]

Manuel del Pino, Monica Musso, Frank Pacard and Angela Pistoia, Torus action on $S^n$ and sign-changing solutions for conformally invariant equations,, \emph{Ann. Sc. Norm. Super. Pisa Cl. Sci.}, 12 (2013), 209.

[12]

Weiyue Ding, On a conformally invariant elliptic equation on $R^n$,, \emph{Comm. Math. Phys.}, 107 (1986), 331.

[13]

Benjamin Dodson, Global well-posedness and scattering for the defocusing, $L^{2}$-critical nonlinear Schrödinger equation when $d\geq3$,, \emph{J. Amer. Math. Soc.}, 25 (2012), 429. doi: 10.1090/S0894-0347-2011-00727-3.

[14]

Roland Donninger and Joachim Krieger, Nonscattering solutions and blowup at infinity for the critical wave equation,, \emph{Math. Ann.}, 357 (2013), 89. doi: 10.1007/s00208-013-0898-1.

[15]

Thomas Duyckaerts, Justin Holmer and Svetlana Roudenko, Scattering for the non-radial 3D cubic nonlinear Schrödinger equation,, \emph{Math. Res. Lett.}, 15 (2008), 1233. doi: 10.4310/MRL.2008.v15.n6.a13.

[16]

Thomas Duyckaerts, Carlos Kenig and Frank Merle, Universality of blow-up profile for small radial type II blow-up solutions of the energy-critical wave equation,, \emph{J. Eur. Math. Soc. (JEMS)}, 13 (2011), 533. doi: 10.4171/JEMS/261.

[17]

Thomas Duyckaerts, Carlos Kenig and Frank Merle, Profiles of bounded radial solutions of the focusing, energy-critical wave equation,, \emph{Geom. Funct. Anal.}, 22 (2012), 639. doi: 10.1007/s00039-012-0174-7.

[18]

Thomas Duyckaerts, Carlos Kenig and Frank Merle, Universality of the blow-up profile for small type II blow-up solutions of the energy-critical wave equation: the nonradial case,, \emph{J. Eur. Math. Soc. (JEMS)}, 14 (2012), 1389. doi: 10.4171/JEMS/336.

[19]

Thomas Duyckaerts, Carlos Kenig and Frank Merle, Classification of radial solutions of the focusing, energy-critical wave equation,, \emph{Cambridge Journal of Mathematics}, 1 (2013), 75. doi: 10.4310/CJM.2013.v1.n1.a3.

[20]

Thomas Duyckaerts, Carlos Kenig and Frank Merle, Scattering for radial, bounded solutions of focusing supercritical wave equations,, \emph{International Mathematics Research Notices}, (2014), 224.

[21]

Thomas Duyckaerts, Carlos Kenig and Frank Merle, Universality of the blow-up profile for small type II blow-up solutions of the energy-critical wave equation: the nonradial case,, Corrected version, (). doi: 10.4171/JEMS/336.

[22]

Thomas Duyckaerts, Carlos Kenig and Frank Merle, Solutions of the focusing, energy-critical wave equation with the compactness property,, Preprint, ().

[23]

Daoyuan Fang, Jian Xie and Thierry Cazenave, Scattering for the focusing energy-subcritical nonlinear Schrödinger equation,, \emph{Sci. China Math.}, 54 (2011), 2037. doi: 10.1007/s11425-011-4283-9.

[24]

Leo Glangetas and Frank Merle, A geometrical approach of existence of blow up solutions in $H^1$ for nonlinear Schrödinger equation,, Preprint, (1995).

[25]

Cristi Guevara, Global behavior of finite energy solutions to the $d$-dimensional focusing nonlinear Schr\"odinger equation,, \emph{Appl. Math. Res. Express. AMRX}, (2014), 177.

[26]

Justin Holmer and Svetlana Roudenko, On blow-up solutions to the 3D cubic nonlinear Schrödinger equation,, \emph{Appl. Math. Res. Express. AMRX}, (2007).

[27]

Carlos E. Kenig and Frank Merle, Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case,, \emph{Invent. Math.}, 166 (2006), 645. doi: 10.1007/s00222-006-0011-4.

[28]

Carlos E. Kenig and Frank Merle, Global well-posedness, scattering and blow-up for the energy-critical focusing non-linear wave equation,, \emph{Acta Math.}, 201 (2008), 147. doi: 10.1007/s11511-008-0031-6.

[29]

Carlos E. Kenig, Andrew Lawrie and Wilhelm Schlag, Relaxation of wave maps exterior to a ball to harmonic maps for all data,, \emph{Geom. Funct. Anal.}, (2014), 610. doi: 10.1007/s00039-014-0262-y.

[30]

Carlos E. Kenig and Frank Merle, Global well-posedness, scattering and blow-up for the energy-critical focusing non-linear wave equation,, \emph{Acta Math.}, 201 (2008), 147. doi: 10.1007/s11511-008-0031-6.

[31]

Sahbi Keraani, On the defect of compactness for the Strichartz estimates of the Schrödinger equations,, \emph{J. Differential Equations}, 175 (2001), 353. doi: 10.1006/jdeq.2000.3951.

[32]

Sahbi Keraani, On the blow up phenomenon of the critical nonlinear Schrödinger equation,, \emph{J. Funct. Anal.}, 235 (2006), 171. doi: 10.1016/j.jfa.2005.10.005.

[33]

Rowan Killip and Monica Visan, The focusing energy-critical nonlinear Schrödinger equation in dimensions five and higher,, \emph{Amer. J. Math.}, 132 (2010), 361. doi: 10.1353/ajm.0.0107.

[34]

Joachim Krieger, Wilhelm Schlag and Daniel Tataru, Slow blow-up solutions for the $H^1(R^3)$ critical focusing semilinear wave equation,, \emph{Duke Math. J.}, 147 (2009), 1. doi: 10.1215/00127094-2009-005.

[35]

Yvan Martel and Frank Merle, A Liouville theorem for the critical generalized Korteweg-de Vries equation,, \emph{J. Math. Pures Appl.}, 79 (2000), 339. doi: 10.1016/S0021-7824(00)00159-8.

[36]

Frank Merle and Pierre Raphael, On universality of blow-up profile for $L^2$ critical nonlinear Schrödinger equation,, \emph{Invent. Math.}, 156 (2004), 565. doi: 10.1007/s00222-003-0346-z.

[37]

Ruipeng Shen, On the energy subcritical, nonlinear wave equation in $\mathbbR^3$ with radial data,, \emph{Anal. PDE}, 6 (2013), 1929. doi: 10.2140/apde.2013.6.1929.

[38]

Per Sjölin, Convergence properties for the Schrödinger equation,, \emph{Rend. Sem. Mat. Fis. Milano}, 57 (1987), 293. doi: 10.1007/BF02925057.

[39]

Jacob Sterbenz and Daniel Tataru, Energy dispersed large data wave maps in $2+1$ dimensions,, \emph{Comm. Math. Phys.}, 298 (2010), 139. doi: 10.1007/s00220-010-1061-4.

[40]

Jacob Sterbenz and Daniel Tataru, Regularity of wave-maps in dimension $2+1$,, \emph{Comm. Math. Phys.}, 298 (2010), 231. doi: 10.1007/s00220-010-1062-3.

[41]

Michael Struwe, Equivariant wave maps in two space dimensions,, \emph{Comm. Pure Appl. Math.}, 56 (2003), 815. doi: 10.1002/cpa.10074.

[42]

Terence Tao, Monica Visan and Xiaoyi Zhang, Minimal-mass blowup solutions of the mass-critical NLS,, \emph{Forum Math.}, 20 (2008), 881. doi: 10.1515/FORUM.2008.042.

[43]

Luis Vega, Schrödinger equations: pointwise convergence to the initial data,, \emph{Proc. Amer. Math. Soc.}, 102 (1988), 874. doi: 10.2307/2047326.

show all references

References:
[1]

Takafumi Akahori and Hayato Nawa, Blowup and scattering problems for the nonlinear Schrödinger equations,, \emph{Kyoto J. Math.}, 53 (2013), 629. doi: 10.1215/21562261-2265914.

[2]

Bulut Aynur, Maximizers for the Strichartz inequalities for the wave equation,, \emph{Differential Integral Equations}, 23 (2010), 1035.

[3]

Hajer Bahouri and Patrick Gérard, High frequency approximation of solutions to critical nonlinear wave equations,, \emph{Amer. J. Math.}, 121 (1999), 131.

[4]

Aynur Bulut, Magdalena Czubak, Dong Li, Nataša Pavlović and Xiaoyi Zhang, Stability and unconditional uniqueness of solutions for energy critical wave equations in high dimensions,, \emph{Comm. Partial Differential Equations}, 38 (2013), 575. doi: 10.1080/03605302.2012.756520.

[5]

Thierry Cazenave and Fred B. Weissler, The Cauchy problem for the critical nonlinear Schrödinger equation in $H^s$,, \emph{Nonlinear Anal.}, 14 (1990), 807. doi: 10.1016/0362-546X(90)90023-A.

[6]

Demetrios Christodoulou and A. Shadi Tahvildar-Zadeh, On the asymptotic behavior of spherically symmetric wave maps,, \emph{Duke Math. J.}, 71 (1993), 31. doi: 10.1215/S0012-7094-93-07103-7.

[7]

Peter Constantin and Jean-Claude Saut, Local smoothing properties of Schrödinger equations,, \emph{Indiana Univ. Math. J.}, 38 (1989), 791. doi: 10.1512/iumj.1989.38.38037.

[8]

Raphaël Côte, Soliton resolution for equivariant wave maps to the sphere,, Preprint, ().

[9]

Manuel del Pino, Personal communication., \quad, ().

[10]

Manuel del Pino, Monica Musso, Frank Pacard and Angela Pistoia, Large energy entire solutions for the Yamabe equation,, \emph{J. Differential Equations}, 251 (2011), 2568. doi: 10.1016/j.jde.2011.03.008.

[11]

Manuel del Pino, Monica Musso, Frank Pacard and Angela Pistoia, Torus action on $S^n$ and sign-changing solutions for conformally invariant equations,, \emph{Ann. Sc. Norm. Super. Pisa Cl. Sci.}, 12 (2013), 209.

[12]

Weiyue Ding, On a conformally invariant elliptic equation on $R^n$,, \emph{Comm. Math. Phys.}, 107 (1986), 331.

[13]

Benjamin Dodson, Global well-posedness and scattering for the defocusing, $L^{2}$-critical nonlinear Schrödinger equation when $d\geq3$,, \emph{J. Amer. Math. Soc.}, 25 (2012), 429. doi: 10.1090/S0894-0347-2011-00727-3.

[14]

Roland Donninger and Joachim Krieger, Nonscattering solutions and blowup at infinity for the critical wave equation,, \emph{Math. Ann.}, 357 (2013), 89. doi: 10.1007/s00208-013-0898-1.

[15]

Thomas Duyckaerts, Justin Holmer and Svetlana Roudenko, Scattering for the non-radial 3D cubic nonlinear Schrödinger equation,, \emph{Math. Res. Lett.}, 15 (2008), 1233. doi: 10.4310/MRL.2008.v15.n6.a13.

[16]

Thomas Duyckaerts, Carlos Kenig and Frank Merle, Universality of blow-up profile for small radial type II blow-up solutions of the energy-critical wave equation,, \emph{J. Eur. Math. Soc. (JEMS)}, 13 (2011), 533. doi: 10.4171/JEMS/261.

[17]

Thomas Duyckaerts, Carlos Kenig and Frank Merle, Profiles of bounded radial solutions of the focusing, energy-critical wave equation,, \emph{Geom. Funct. Anal.}, 22 (2012), 639. doi: 10.1007/s00039-012-0174-7.

[18]

Thomas Duyckaerts, Carlos Kenig and Frank Merle, Universality of the blow-up profile for small type II blow-up solutions of the energy-critical wave equation: the nonradial case,, \emph{J. Eur. Math. Soc. (JEMS)}, 14 (2012), 1389. doi: 10.4171/JEMS/336.

[19]

Thomas Duyckaerts, Carlos Kenig and Frank Merle, Classification of radial solutions of the focusing, energy-critical wave equation,, \emph{Cambridge Journal of Mathematics}, 1 (2013), 75. doi: 10.4310/CJM.2013.v1.n1.a3.

[20]

Thomas Duyckaerts, Carlos Kenig and Frank Merle, Scattering for radial, bounded solutions of focusing supercritical wave equations,, \emph{International Mathematics Research Notices}, (2014), 224.

[21]

Thomas Duyckaerts, Carlos Kenig and Frank Merle, Universality of the blow-up profile for small type II blow-up solutions of the energy-critical wave equation: the nonradial case,, Corrected version, (). doi: 10.4171/JEMS/336.

[22]

Thomas Duyckaerts, Carlos Kenig and Frank Merle, Solutions of the focusing, energy-critical wave equation with the compactness property,, Preprint, ().

[23]

Daoyuan Fang, Jian Xie and Thierry Cazenave, Scattering for the focusing energy-subcritical nonlinear Schrödinger equation,, \emph{Sci. China Math.}, 54 (2011), 2037. doi: 10.1007/s11425-011-4283-9.

[24]

Leo Glangetas and Frank Merle, A geometrical approach of existence of blow up solutions in $H^1$ for nonlinear Schrödinger equation,, Preprint, (1995).

[25]

Cristi Guevara, Global behavior of finite energy solutions to the $d$-dimensional focusing nonlinear Schr\"odinger equation,, \emph{Appl. Math. Res. Express. AMRX}, (2014), 177.

[26]

Justin Holmer and Svetlana Roudenko, On blow-up solutions to the 3D cubic nonlinear Schrödinger equation,, \emph{Appl. Math. Res. Express. AMRX}, (2007).

[27]

Carlos E. Kenig and Frank Merle, Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case,, \emph{Invent. Math.}, 166 (2006), 645. doi: 10.1007/s00222-006-0011-4.

[28]

Carlos E. Kenig and Frank Merle, Global well-posedness, scattering and blow-up for the energy-critical focusing non-linear wave equation,, \emph{Acta Math.}, 201 (2008), 147. doi: 10.1007/s11511-008-0031-6.

[29]

Carlos E. Kenig, Andrew Lawrie and Wilhelm Schlag, Relaxation of wave maps exterior to a ball to harmonic maps for all data,, \emph{Geom. Funct. Anal.}, (2014), 610. doi: 10.1007/s00039-014-0262-y.

[30]

Carlos E. Kenig and Frank Merle, Global well-posedness, scattering and blow-up for the energy-critical focusing non-linear wave equation,, \emph{Acta Math.}, 201 (2008), 147. doi: 10.1007/s11511-008-0031-6.

[31]

Sahbi Keraani, On the defect of compactness for the Strichartz estimates of the Schrödinger equations,, \emph{J. Differential Equations}, 175 (2001), 353. doi: 10.1006/jdeq.2000.3951.

[32]

Sahbi Keraani, On the blow up phenomenon of the critical nonlinear Schrödinger equation,, \emph{J. Funct. Anal.}, 235 (2006), 171. doi: 10.1016/j.jfa.2005.10.005.

[33]

Rowan Killip and Monica Visan, The focusing energy-critical nonlinear Schrödinger equation in dimensions five and higher,, \emph{Amer. J. Math.}, 132 (2010), 361. doi: 10.1353/ajm.0.0107.

[34]

Joachim Krieger, Wilhelm Schlag and Daniel Tataru, Slow blow-up solutions for the $H^1(R^3)$ critical focusing semilinear wave equation,, \emph{Duke Math. J.}, 147 (2009), 1. doi: 10.1215/00127094-2009-005.

[35]

Yvan Martel and Frank Merle, A Liouville theorem for the critical generalized Korteweg-de Vries equation,, \emph{J. Math. Pures Appl.}, 79 (2000), 339. doi: 10.1016/S0021-7824(00)00159-8.

[36]

Frank Merle and Pierre Raphael, On universality of blow-up profile for $L^2$ critical nonlinear Schrödinger equation,, \emph{Invent. Math.}, 156 (2004), 565. doi: 10.1007/s00222-003-0346-z.

[37]

Ruipeng Shen, On the energy subcritical, nonlinear wave equation in $\mathbbR^3$ with radial data,, \emph{Anal. PDE}, 6 (2013), 1929. doi: 10.2140/apde.2013.6.1929.

[38]

Per Sjölin, Convergence properties for the Schrödinger equation,, \emph{Rend. Sem. Mat. Fis. Milano}, 57 (1987), 293. doi: 10.1007/BF02925057.

[39]

Jacob Sterbenz and Daniel Tataru, Energy dispersed large data wave maps in $2+1$ dimensions,, \emph{Comm. Math. Phys.}, 298 (2010), 139. doi: 10.1007/s00220-010-1061-4.

[40]

Jacob Sterbenz and Daniel Tataru, Regularity of wave-maps in dimension $2+1$,, \emph{Comm. Math. Phys.}, 298 (2010), 231. doi: 10.1007/s00220-010-1062-3.

[41]

Michael Struwe, Equivariant wave maps in two space dimensions,, \emph{Comm. Pure Appl. Math.}, 56 (2003), 815. doi: 10.1002/cpa.10074.

[42]

Terence Tao, Monica Visan and Xiaoyi Zhang, Minimal-mass blowup solutions of the mass-critical NLS,, \emph{Forum Math.}, 20 (2008), 881. doi: 10.1515/FORUM.2008.042.

[43]

Luis Vega, Schrödinger equations: pointwise convergence to the initial data,, \emph{Proc. Amer. Math. Soc.}, 102 (1988), 874. doi: 10.2307/2047326.

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