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Uniform global existence and convergence of Euler-Maxwell systems with small parameters
Global dynamics above the ground state for the energy-critical Schrödinger equation with radial data
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 |
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. |
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. |
[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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