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Volumn 39, Issue 1, 2007, Pages 34-56

On the blowup for the L2-Critical focusing nonlinear Schrödinger equation in higher dimensions below the energy class

Author keywords

Blowup; I method; Nonlinear Schr dinger equation

Indexed keywords

BLOWUP; ENERGY CLASS; ENERGY SPACE;

EID: 39449130810     PISSN: 00361410     EISSN: None     Source Type: Journal    
DOI: 10.1137/060663969     Document Type: Article
Times cited : (16)

References (42)
  • 1
    • 55549107108 scopus 로고    scopus 로고
    • 2-Critical Nonlinear Schrödinger Equation
    • preprint
    • 2-Critical Nonlinear Schrödinger Equation, preprint.
    • BEGOUT, P.1    VARGAS, A.2
  • 2
    • 0000193318 scopus 로고    scopus 로고
    • Refinements of Strichartz inequality and applications to 2D-NLS with critical nonlinearity
    • J. BOURGAIN, Refinements of Strichartz inequality and applications to 2D-NLS with critical nonlinearity, Internat. Math. Res. Notices, 5 (1998), pp. 253-283.
    • (1998) Internat. Math. Res. Notices , vol.5 , pp. 253-283
    • BOURGAIN, J.1
  • 3
    • 0000046869 scopus 로고    scopus 로고
    • Scattering in the energy space and below for 3D NLS
    • J. BOURGAIN, Scattering in the energy space and below for 3D NLS, J. Anal. Math., 75 (1998), pp. 267-297.
    • (1998) J. Anal. Math , vol.75 , pp. 267-297
    • BOURGAIN, J.1
  • 5
    • 23244446904 scopus 로고
    • Critical nonlinear Schrödinger Equation
    • T. CAZENAVB AND F. B. WEISSLER, Critical nonlinear Schrödinger Equation, Nonlinear Anal., 14 (1990), pp. 807-836.
    • (1990) Nonlinear Anal , vol.14 , pp. 807-836
    • CAZENAVB, T.1    WEISSLER, F.B.2
  • 7
    • 0002645561 scopus 로고
    • Dispersion of small amplitude solutions of the generalized Korteweg-de Vries equation
    • M. CHRIST AND M. WEINSTEIN, Dispersion of small amplitude solutions of the generalized Korteweg-de Vries equation, J. Funct. Anal., 100 (1991), pp. 87-109.
    • (1991) J. Funct. Anal , vol.100 , pp. 87-109
    • CHRIST, M.1    WEINSTEIN, M.2
  • 8
    • 84966238915 scopus 로고
    • On commutators of singular integrals and bilinear singular integrals
    • R. R. COIFMAN AND Y. MEYER, On commutators of singular integrals and bilinear singular integrals, Trans. Amer. Math. Soc., 212 (1975), pp. 315-331.
    • (1975) Trans. Amer. Math. Soc , vol.212 , pp. 315-331
    • COIFMAN, R.R.1    MEYER, Y.2
  • 9
    • 39449120501 scopus 로고    scopus 로고
    • R. R. COIFMAN AND Y. MEYER, Ondelettes et opérateurs. III, Opérateurs multilinéaires, Actualites Math., Hermann, Paris, 1991.
    • R. R. COIFMAN AND Y. MEYER, Ondelettes et opérateurs. III, Opérateurs multilinéaires, Actualites Math., Hermann, Paris, 1991.
  • 10
    • 0036771238 scopus 로고    scopus 로고
    • Almost conservation laws and global rough solutions to a nonlinear Schrödinger equation
    • J. COLLIANDER, M. KEEL, G. STAFFILANI, H. TAKAOKA, AND T. TAO, Almost conservation laws and global rough solutions to a nonlinear Schrödinger equation, Math. Res. Lett., 9 (2002), pp. 659-682.
    • (2002) Math. Res. Lett , vol.9 , pp. 659-682
    • COLLIANDER, J.1    KEEL, M.2    STAFFILANI, G.3    TAKAOKA, H.4    TAO, T.5
  • 11
    • 0037254091 scopus 로고    scopus 로고
    • Polynomial upper bounds for the orbital instability of the 1D cubic NLS below the energy norm
    • J. COLLIANDER, M. KEEL, G. STAFFILANI, H. TAKAOKA, AND T. TAO, Polynomial upper bounds for the orbital instability of the 1D cubic NLS below the energy norm, Discrete Contin. Dyn. Syst., 9 (2003), pp. 31-54.
    • (2003) Discrete Contin. Dyn. Syst , vol.9 , pp. 31-54
    • COLLIANDER, J.1    KEEL, M.2    STAFFILANI, G.3    TAKAOKA, H.4    TAO, T.5
  • 12
    • 23244434956 scopus 로고    scopus 로고
    • Polynomial upper bounds for the instability of the nonlinear Schrödinger equation below the energy norm
    • J. COLLIANDER, M. KEEL, G. STAFFILANI, H. TAKAOKA, AND T. TAO, Polynomial upper bounds for the instability of the nonlinear Schrödinger equation below the energy norm, Commun. Pure Appl. Anal., 2 (2003), pp. 33-50.
    • (2003) Commun. Pure Appl. Anal , vol.2 , pp. 33-50
    • COLLIANDER, J.1    KEEL, M.2    STAFFILANI, G.3    TAKAOKA, H.4    TAO, T.5
  • 15
    • 84996132207 scopus 로고    scopus 로고
    • Description du defaut de compacite de l'injection de Sobolev
    • P. GERARD, Description du defaut de compacite de l'injection de Sobolev, ESAIM Control Optim. Calc. Var., 3 (1998), pp. 213-233.
    • (1998) ESAIM Control Optim. Calc. Var , vol.3 , pp. 213-233
    • GERARD, P.1
  • 16
    • 0000555922 scopus 로고
    • Scattering theory in the energy space for a class of nonlinear Schrödinger equations
    • J. GINIBRE AND G. VELO, Scattering theory in the energy space for a class of nonlinear Schrödinger equations, J. Math. Pure. Appl., 64 (1985), pp. 363-401.
    • (1985) J. Math. Pure. Appl , vol.64 , pp. 363-401
    • GINIBRE, J.1    VELO, G.2
  • 17
    • 0001123370 scopus 로고
    • Smoothing properties and retarded estimates for some dispersive evolution equations
    • J. GINIBRE AND G. VELO, Smoothing properties and retarded estimates for some dispersive evolution equations, Comm. Math. Phys., 144 (1992), pp. 163-188.
    • (1992) Comm. Math. Phys , vol.144 , pp. 163-188
    • GINIBRE, J.1    VELO, G.2
  • 18
    • 36749114276 scopus 로고
    • On the blowing up of solution to the Cauchy problem for nonlinear Schrödinger operators
    • R. T. GLASSEY, On the blowing up of solution to the Cauchy problem for nonlinear Schrödinger operators, J. Math. Phys., 8 (1977), pp. 1794-1797.
    • (1977) J. Math. Phys , vol.8 , pp. 1794-1797
    • GLASSEY, R.T.1
  • 19
    • 0039511979 scopus 로고
    • Remarks on the scattering problem for nonlinear Schrödinger equations
    • Differential Equations and Mathematical Physics Birmingham, AL, Springer, Berlin
    • N. HAYASHI AND Y. TSUTSUMI, Remarks on the scattering problem for nonlinear Schrödinger equations, in Differential Equations and Mathematical Physics (Birmingham, AL, 1986), Lecture Notes in Math. 1285, Springer, Berlin, 1987, pp. 162-168.
    • (1986) Lecture Notes in Math , vol.1285 , pp. 162-168
    • HAYASHI, N.1    TSUTSUMI, Y.2
  • 20
    • 28444450137 scopus 로고    scopus 로고
    • Blowup theory for the critical nonlinear Schrödinger equations revisited
    • T. HMIDI AND S. KERAANI, Blowup theory for the critical nonlinear Schrödinger equations revisited, Int. Math. Res. Not., 46 (2005), pp. 2815-2828.
    • (2005) Int. Math. Res. Not , vol.46 , pp. 2815-2828
    • HMIDI, T.1    KERAANI, S.2
  • 21
    • 39449124672 scopus 로고    scopus 로고
    • 2-Critical Nonlinear Schrödinger Equations
    • preprint
    • 2-Critical Nonlinear Schrödinger Equations, preprint.
    • HMIDI, T.1    KERAANI, S.2
  • 22
    • 0001404168 scopus 로고
    • On nonlinear Schrödinger equations
    • T. KATO, On nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Phys. Théor., 46 (1987), pp. 113-129.
    • (1987) Ann. Inst. H. Poincaré Phys. Théor , vol.46 , pp. 113-129
    • KATO, T.1
  • 24
    • 0001138601 scopus 로고    scopus 로고
    • Endpoint Strichartz estimates
    • M. KEEL AND T. TAO, Endpoint Strichartz estimates, Amer. Math. J., 120 (1998), pp. 955-980.
    • (1998) Amer. Math. J , vol.120 , pp. 955-980
    • KEEL, M.1    TAO, T.2
  • 25
    • 33645945606 scopus 로고    scopus 로고
    • On the blow up phenomenon of the critical nonlinear Schrödinger equation
    • S. KERAANI, On the blow up phenomenon of the critical nonlinear Schrödinger equation, J. Funct. Anal., 235 (2006), pp. 171-192.
    • (2006) J. Funct. Anal , vol.235 , pp. 171-192
    • KERAANI, S.1
  • 26
    • 34249974409 scopus 로고    scopus 로고
    • n, Arch. Rational Mech. Anal., 105 (1989), pp. 243-266.
    • n, Arch. Rational Mech. Anal., 105 (1989), pp. 243-266.
  • 27
    • 0001304768 scopus 로고
    • Arch. Rational Mech. Anal, 99
    • n, Arch. Rational Mech. Anal., 99 (1987), pp. 115-145.
    • (1987) n , pp. 115-145
    • MCLEOD, K.1    SERRIN, J.2
  • 28
    • 84974001368 scopus 로고
    • Determination of blow-up solutions with minimal mass for nonlinear Schrödinger equations with critical power
    • F. MERLE, Determination of blow-up solutions with minimal mass for nonlinear Schrödinger equations with critical power, Duke Math. J., 69 (1993), pp. 427-454.
    • (1993) Duke Math. J , vol.69 , pp. 427-454
    • MERLE, F.1
  • 29
    • 20444437961 scopus 로고    scopus 로고
    • The blow-up dynamic and upper bound on the blow-up rate for critical nonlinear Schrödinger equation
    • F. MERLE AND P. RAPHAEL, The blow-up dynamic and upper bound on the blow-up rate for critical nonlinear Schrödinger equation, Ann. of Math. (2), 161 (2005), pp. 157-222.
    • (2005) Ann. of Math. (2) , vol.161 , pp. 157-222
    • MERLE, F.1    RAPHAEL, P.2
  • 30
    • 12444281836 scopus 로고    scopus 로고
    • Profiles and quantization of the blow up mass for critical nonlinear Schrödinger equation
    • F. MERLE AND P. RAPHAEL, Profiles and quantization of the blow up mass for critical nonlinear Schrödinger equation, Comm. Math. Phys., 253 (2005), pp. 675-704.
    • (2005) Comm. Math. Phys , vol.253 , pp. 675-704
    • MERLE, F.1    RAPHAEL, P.2
  • 31
    • 0000704928 scopus 로고
    • 2- concentration of blow-up solutions for the nonlinear Schrödinger equation with the critical power nonlinearity
    • 2- concentration of blow-up solutions for the nonlinear Schrödinger equation with the critical power nonlinearity, J. Differential Equations, 84 (1990), pp. 205-214.
    • (1990) J. Differential Equations , vol.84 , pp. 205-214
    • MERLE, F.1    TSUTSUMI, Y.2
  • 32
    • 0347852485 scopus 로고    scopus 로고
    • On the generalized Korteweg-de Vries-type equations
    • G. STAFFILANI, On the generalized Korteweg-de Vries-type equations, Differential Integral Equations, 10 (1997), pp. 777-796.
    • (1997) Differential Integral Equations , vol.10 , pp. 777-796
    • STAFFILANI, G.1
  • 33
    • 84972553620 scopus 로고
    • Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations
    • R. S. STRICHARTZ, Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke Math. J., 44 (1977), pp. 705-714.
    • (1977) Duke Math. J , vol.44 , pp. 705-714
    • STRICHARTZ, R.S.1
  • 34
    • 15944416090 scopus 로고    scopus 로고
    • On the asymptotic behavior of large radial data for a focusing non-linear Schrödinger equation
    • T. TAO, On the asymptotic behavior of large radial data for a focusing non-linear Schrödinger equation, Dyn. Partial Differ. Equ., 1 (2004), pp. 1-48.
    • (2004) Dyn. Partial Differ. Equ , vol.1 , pp. 1-48
    • TAO, T.1
  • 35
    • 0010786492 scopus 로고    scopus 로고
    • Tools for PDE. Pseudodifferential Operators, Paradifferential Operators, and Layer Potentials, Math
    • AMS, Providence, RI
    • M. E. TAYLOR, Tools for PDE. Pseudodifferential Operators, Paradifferential Operators, and Layer Potentials, Math. Surveys Monogr. 81, AMS, Providence, RI, 2000.
    • (2000) Surveys Monogr , vol.81
    • TAYLOR, M.E.1
  • 36
    • 33751011022 scopus 로고    scopus 로고
    • Moss concentration phenomenon for the quintic nonlinear Schrödinger equation in one dimension
    • N. TZIRAKIS, Moss concentration phenomenon for the quintic nonlinear Schrödinger equation in one dimension, SIAM J. Math. Anal., 37 (2006), pp. 1923-1946.
    • (2006) SIAM J. Math. Anal , vol.37 , pp. 1923-1946
    • TZIRAKIS, N.1
  • 38
    • 34547304016 scopus 로고    scopus 로고
    • The defocusing energy-critical nonlinear Schrödinger equation in higher dimensions
    • to appear
    • M. VISAN, The defocusing energy-critical nonlinear Schrödinger equation in higher dimensions, Duke Math. J., to appear.
    • Duke Math. J
    • VISAN, M.1
  • 39
    • 0041473959 scopus 로고
    • Nonlinear Schrödinger equations and sharp interpolation estimates
    • M. I. WEINSTEIN, Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys., 87 (1982/1983), pp. 567-576.
    • (1982) Comm. Math. Phys , vol.87 , pp. 567-576
    • WEINSTEIN, M.I.1
  • 40
    • 0000686130 scopus 로고
    • Modulational stability of ground states of nonlinear Schrödinger equations
    • M. I. WEINSTEIN, Modulational stability of ground states of nonlinear Schrödinger equations, SIAM J. Math. Anal., 16 (1985), pp. 472-491.
    • (1985) SIAM J. Math. Anal , vol.16 , pp. 472-491
    • WEINSTEIN, M.I.1
  • 41
    • 84990553584 scopus 로고
    • Lyapunov stability of ground states of nonlinear Schrödinger equations
    • M. I. WEINSTEIN, Lyapunov stability of ground states of nonlinear Schrödinger equations, Comm. Pure Appl. Math., 39 (1986), pp. 51-68.
    • (1986) Comm. Pure Appl. Math , vol.39 , pp. 51-68
    • WEINSTEIN, M.I.1
  • 42
    • 0000242295 scopus 로고
    • Quasi-classical theory for three-dimensional wave collapse
    • V. E. ZAKHAROV AND E. A. KUZNETSOV, Quasi-classical theory for three-dimensional wave collapse, Soviet Phys. JETP, 64 (1986), pp. 773-380.
    • (1986) Soviet Phys. JETP , vol.64 , pp. 773-380
    • ZAKHAROV, V.E.1    KUZNETSOV, E.A.2


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