메뉴 건너뛰기




Volumn 135, Issue 4, 2013, Pages 967-1017

The instability of Bourgain-Wang solutions for the L2 critical NLS

Author keywords

[No Author keywords available]

Indexed keywords


EID: 84880789513     PISSN: 00029327     EISSN: 10806377     Source Type: Journal    
DOI: 10.1353/ajm.2013.0033     Document Type: Article
Times cited : (64)

References (51)
  • 1
  • 2
    • 0000246714 scopus 로고    scopus 로고
    • Construction of blowup solutions for the nonlinear Schrödinger equation with critical nonlinearity
    • J. Bourgain and W. Wang, Construction of blowup solutions for the nonlinear Schrödinger equation with critical nonlinearity, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 25 (1997), no. 1-2, 197-215.
    • (1997) Ann. Scuola Norm. Sup. Pisa Cl. Sci , vol.25 , Issue.1-2 , pp. 197-215
    • Bourgain, J.1    Wang, W.2
  • 3
    • 5444256656 scopus 로고    scopus 로고
    • Semilinear Schrödinger Equations
    • New York University Courant Institute of Mathematical Sciences, New York
    • T. Cazenave, Semilinear Schrödinger Equations, Courant Lect. Notes Math., vol. 10, New York University Courant Institute of Mathematical Sciences, New York, 2003.
    • (2003) Courant Lect. Notes Math , vol.10
    • Cazenave, T.1
  • 4
    • 0011576999 scopus 로고
    • Some remarks on the nonlinear Schrödinger equation in the critical case
    • (Washington, DC, 1987), Lecture Notes in Math, Springer-Verlag, Berlin
    • T. Cazenave and F. B. Weissler, Some remarks on the nonlinear Schrödinger equation in the critical case, Nonlinear Semigroups, Partial Differential Equations and Attractors (Washington, DC, 1987), Lecture Notes in Math., vol. 1394, Springer-Verlag, Berlin, 1989, pp. 18-29.
    • (1989) Nonlinear Semigroups, Partial Differential Equations and Attractors , vol.1394 , pp. 18-29
    • Cazenave, T.1    Weissler, F.B.2
  • 7
    • 69549133355 scopus 로고    scopus 로고
    • Dynamics of threshold solutions for energy-critical wave equation
    • Art ID rpn002, 67
    • T. Duyckaerts and F. Merle, Dynamics of threshold solutions for energy-critical wave equation, Int. Math. Res. Pap. IMRP (2008), Art ID rpn002, 67.
    • (2008) Int. Math. Res. Pap. IMRP
    • Duyckaerts, T.1    Merle, F.2
  • 8
    • 63949086654 scopus 로고    scopus 로고
    • Dynamic of threshold solutions for energy-critical NLS
    • T. Duyckaerts and F. Merle, Dynamic of threshold solutions for energy-critical NLS, Geom. Funct. Anal. 18 (2009), no. 6, 1787-1840.
    • (2009) Geom. Funct. Anal , vol.18 , Issue.6 , pp. 1787-1840
    • Duyckaerts, T.1    Merle, F.2
  • 9
    • 77951143619 scopus 로고    scopus 로고
    • Threshold solutions for the focusing 3D cubic Schrödinger equation
    • T. Duyckaerts and S. Roudenko, Threshold solutions for the focusing 3D cubic Schrödinger equation, Rev. Mat. Iberoam. 26 (2010), no. 1, 1-56.
    • (2010) Rev. Mat. Iberoam , vol.26 , Issue.1 , pp. 1-56
    • Duyckaerts, T.1    Roudenko, S.2
  • 10
    • 34250271532 scopus 로고
    • Symmetry and related properties via the maximum principle
    • B. Gidas, W. M. Ni, and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 (1979), no. 3, 209-243.
    • (1979) Comm. Math. Phys , vol.68 , Issue.3 , pp. 209-243
    • Gidas, B.1    Ni, W.M.2    Nirenberg, L.3
  • 11
    • 49249148441 scopus 로고
    • On a class of nonlinear Schrödinger equations I. the Cauchy Problem
    • J. Ginibre and G. Velo, On a class of nonlinear Schrödinger equations. I. The Cauchy problem, general case, J. Funct. Anal. 32 (1979), no. 1, 1-32.
    • (1979) General Case, J. Funct. Anal , vol.32 , Issue.1 , pp. 1-32
    • Ginibre, J.1    Velo, G.2
  • 13
    • 70450169280 scopus 로고    scopus 로고
    • Characterization of minimal-mass blowup solutions to the focusing mass-critical NLS
    • R. Killip, D. Li, M. Visan, and X. Zhang, Characterization of minimal-mass blowup solutions to the focusing mass-critical NLS, SIAM J. Math. Anal. 41 (2009), no. 1, 219-236.
    • (2009) SIAM J. Math. Anal , vol.41 , Issue.1 , pp. 219-236
    • Killip, R.1    Li, D.2    Visan, M.3    Zhang, X.4
  • 14
    • 74549167807 scopus 로고    scopus 로고
    • The cubic nonlinear Schrödinger equation in two dimensions with radial data
    • R. Killip, T. Tao, and M. Visan, The cubic nonlinear Schrödinger equation in two dimensions with radial data, J. Eur. Math. Soc. (JEMS) 11 (2009), no. 6, 1203-1258.
    • (2009) J. Eur. Math. Soc. (JEMS) , vol.11 , Issue.6 , pp. 1203-1258
    • Killip, R.1    Tao, T.2    Visan, M.3
  • 15
    • 33749459423 scopus 로고    scopus 로고
    • Stable manifolds for all monic supercritical focusing nonlinear Schrödinger equations in one dimension
    • J. Krieger and W. Schlag, Stable manifolds for all monic supercritical focusing nonlinear Schrödinger equations in one dimension, J. Amer. Math. Soc. 19 (2006), no. 4, 815-920.
    • (2006) J. Amer. Math. Soc , vol.19 , Issue.4 , pp. 815-920
    • Krieger, J.1    Schlag, W.2
  • 16
    • 34250849129 scopus 로고    scopus 로고
    • On the focusing critical semi-linear wave equation
    • J. Krieger and W. Schlag, On the focusing critical semi-linear wave equation, Amer. J.Math. 129 (2007), no. 3, 843-913.
    • (2007) Amer. J.Math , vol.129 , Issue.3 , pp. 843-913
    • Krieger, J.1    Schlag, W.2
  • 17
    • 70350005122 scopus 로고    scopus 로고
    • Non-generic blow-up solutions for the critical focusing NLS in 1-D
    • J. Krieger and W. Schlag, Non-generic blow-up solutions for the critical focusing NLS in 1-D, J. Eur. Math. Soc. (JEMS) 11 (2009), no. 1, 1-125.
    • (2009) J. Eur. Math. Soc. (JEMS) , vol.11 , Issue.1 , pp. 1-125
    • Krieger, J.1    Schlag, W.2
  • 18
    • 38349190717 scopus 로고    scopus 로고
    • Renormalization and blow up for charge one equivariant critical wave maps
    • J. Krieger, W. Schlag, and D. Tataru, Renormalization and blow up for charge one equivariant critical wave maps, Invent. Math. 171 (2008), no. 3, 543-615.
    • (2008) Invent. Math , vol.171 , Issue.3 , pp. 543-615
    • Krieger, J.1    Schlag, W.2    Tataru, D.3
  • 19
    • 67349201747 scopus 로고    scopus 로고
    • Renormalization and blow up for the critical Yang-Mills problem
    • J. Krieger and W. Schlag, Renormalization and blow up for the critical Yang-Mills problem, Adv. Math. 221 (2009), no. 5, 1445-1521.
    • (2009) Adv. Math , vol.221 , Issue.5 , pp. 1445-1521
    • Krieger, J.1    Schlag, W.2
  • 20
    • 67349141836 scopus 로고    scopus 로고
    • 3) critical focusing semilinear wave equation
    • 3) critical focusing semilinear wave equation, Duke Math. J. 147 (2009), no. 1, 1-53.
    • (2009) Duke Math. J , vol.147 , Issue.1 , pp. 1-53
    • Krieger, J.1    Schlag, W.2
  • 22
    • 0000893351 scopus 로고
    • Rate of blowup for solutions of the nonlinear Schrödinger equation at critical dimension
    • M. J. Landman, G. C. Papanicolaou, C. Sulem, and P.-L. Sulem, Rate of blowup for solutions of the nonlinear Schrödinger equation at critical dimension, Phys. Rev. A (3) 38 (1988), no. 8, 3837-3843.
    • (1988) Phys. Rev. a , vol.38 , Issue.3-8 , pp. 3837-3843
    • Landman, M.J.1    Papanicolaou, G.C.2    Sulem, C.3    Sulem, P.-L.4
  • 23
    • 84863045199 scopus 로고    scopus 로고
    • On the rigidity of solitary waves for the focusing mass-critical NLS in dimensions d ≥ 2
    • D. Li and X. Zhang, On the rigidity of solitary waves for the focusing mass-critical NLS in dimensions d ≥ 2, Sci. China Math. 55 (2012), no. 2, 385-434.
    • (2012) Sci. China Math , vol.55 , Issue.2 , pp. 385-434
    • Li, D.1    Zhang, X.2
  • 24
    • 27544509563 scopus 로고    scopus 로고
    • Asymptotic N-soliton-like solutions of the subcritical and critical generalized Korteweg-de Vries equations
    • Y. Martel, Asymptotic N-soliton-like solutions of the subcritical and critical generalized Korteweg-de Vries equations, Amer. J. Math. 127 (2005), no. 5, 1103-1140.
    • (2005) Amer. J. Math , vol.127 , Issue.5 , pp. 1103-1140
    • Martel, Y.1
  • 25
    • 58149329097 scopus 로고    scopus 로고
    • Classification of type I and type II behaviors for a supercritical nonlinear heat equation
    • H. Matano and F. Merle, Classification of type I and type II behaviors for a supercritical nonlinear heat equation, J. Funct. Anal. 256 (2009), no. 4, 992-1064.
    • (2009) J. Funct. Anal , vol.256 , Issue.4 , pp. 992-1064
    • Matano, H.1    Merle, F.2
  • 26
    • 0001337201 scopus 로고
    • Construction of solutions with exactly k blow-up points for the Schrödinger equation with critical nonlinearity
    • F. Merle, Construction of solutions with exactly k blow-up points for the Schrödinger equation with critical nonlinearity, Comm. Math. Phys. 129 (1990), no. 2, 223-240.
    • (1990) Comm. Math. Phys , vol.129 , Issue.2 , pp. 223-240
    • Merle, F.1
  • 27
    • 84990553632 scopus 로고
    • On uniqueness and continuation properties after blow-up time of self-similar solutions of nonlinear Schrödinger equation with critical exponent and critical mass
    • J. Krieger and W. Schlag, On uniqueness and continuation properties after blow-up time of self-similar solutions of nonlinear Schrödinger equation with critical exponent and critical mass, Comm. Pure Appl. Math. 45 (1992), no. 2, 203-254.
    • (1992) Comm. Pure Appl. Math , vol.45 , Issue.2 , pp. 203-254
    • Krieger, J.1    Schlag, W.2
  • 28
    • 84974001368 scopus 로고
    • Determination of blow-up solutions with minimal mass for nonlinear Schrödinger equations with critical power
    • J. Krieger and W. Schlag, Determination of blow-up solutions with minimal mass for nonlinear Schrödinger equations with critical power, Duke Math. J. 69 (1993), no. 2, 427-454.
    • (1993) Duke Math. J , vol.69 , Issue.2 , pp. 427-454
    • Krieger, J.1    Schlag, W.2
  • 30
    • 0041350485 scopus 로고    scopus 로고
    • Sharp upper bound on the blow-up rate for the critical nonlinear Schrödinger equation
    • F. Merle and P. Raphaël, Sharp upper bound on the blow-up rate for the critical nonlinear Schrödinger equation, Geom. Funct. Anal. 13 (2003), no. 3, 591-642.
    • (2003) Geom. Funct. Anal , vol.13 , Issue.3 , pp. 591-642
    • Merle, F.1    Raphaël, P.2
  • 31
    • 2942586742 scopus 로고    scopus 로고
    • On universality of blow-up profile for L2 critical nonlinear Schrödinger equation
    • F. Merle and P. Raphaël, On universality of blow-up profile for L2 critical nonlinear Schrödinger equation, Invent. Math. 156 (2004), no. 3, 565-672.
    • (2004) Invent. Math , vol.156 , Issue.3 , pp. 565-672
    • Merle, F.1    Raphaël, P.2
  • 32
    • 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. Raphaël, The blow-up dynamic and upper bound on the blow-up rate for critical nonlinear Schrödinger equation, Ann. of Math. (2) 161 (2005), no. 1, 157-222.
    • (2005) Ann. of Math) , vol.161 , Issue.1-2 , pp. 157-222
    • Merle, F.1    Raphaël, P.2
  • 33
    • 34547283966 scopus 로고    scopus 로고
    • On one blow up point solutions to the critical nonlinear Schrödinger equation
    • F. Merle and P. Raphaël, On one blow up point solutions to the critical nonlinear Schrödinger equation, J. Hyperbolic Differ. Equ. 2 (2005), no. 4, 919-962.
    • (2005) J. Hyperbolic Differ. Equ , vol.2 , Issue.4 , pp. 919-962
    • Merle, F.1    Raphaël, P.2
  • 34
    • 12444281836 scopus 로고    scopus 로고
    • Profiles and quantization of the blow up mass for critical nonlinear Schrödinger equation
    • F. Merle and P. Raphaël, Profiles and quantization of the blow up mass for critical nonlinear Schrödinger equation, Comm. Math. Phys. 253 (2005), no. 3, 675-704.
    • (2005) Comm. Math. Phys , vol.253 , Issue.3 , pp. 675-704
    • Merle, F.1    Raphaël, P.2
  • 35
    • 30644464564 scopus 로고    scopus 로고
    • On a sharp lower bound on the blow-up rate for the L2 critical nonlinear Schrödinger equation
    • F. Merle and P. Raphaël, On a sharp lower bound on the blow-up rate for the L2 critical nonlinear Schrödinger equation, J. Amer. Math. Soc. 19 (2006), no. 1, 37-90.
    • (2006) J. Amer. Math. Soc , vol.19 , Issue.1 , pp. 37-90
    • Merle, F.1    Raphaël, P.2
  • 36
    • 50049127248 scopus 로고    scopus 로고
    • Blow up of the critical norm for some radial L2 super critical nonlinear Schrödinger equations
    • F. Merle and P. Raphaël, Blow up of the critical norm for some radial L2 super critical nonlinear Schrödinger equations, Amer. J. Math. 130 (2008), no. 4, 945-978.
    • (2008) Amer. J. Math , vol.130 , Issue.4 , pp. 945-978
    • Merle, F.1    Raphaël, P.2
  • 37
    • 78651519686 scopus 로고    scopus 로고
    • Global dynamics above the ground state energy for the focusing nonlinear Klein-Gordon equation
    • K. Nakanishi and W. Schlag, Global dynamics above the ground state energy for the focusing nonlinear Klein-Gordon equation, J. Differential Equations 250 (2011), no. 5, 2299-2333.
    • (2011) J. Differential Equations , vol.250 , Issue.5 , pp. 2299-2333
    • Nakanishi, K.1    Schlag, W.2
  • 38
    • 85027941065 scopus 로고    scopus 로고
    • Global dynamics above the ground state energy for the cubic NLS equation in 3D
    • K. Nakanishi and W. Schlag, Global dynamics above the ground state energy for the cubic NLS equation in 3D, Calc. Var. Partial Differential Equations 44 (2012), no. 1-2, 1-45.
    • (2012) Calc. Var. Partial Differential Equations , vol.44 , Issue.1-2 , pp. 1-45
    • Nakanishi, K.1    Schlag, W.2
  • 39
    • 0035537074 scopus 로고    scopus 로고
    • On the formation of singularities in solutions of the critical nonlinear Schrödinger equation
    • G. Perelman, On the formation of singularities in solutions of the critical nonlinear Schrödinger equation, Ann. Henri Poincaré 2 (2001), no. 4, 605-673.
    • (2001) Ann. Henri Poincaré , vol.2 , Issue.4 , pp. 605-673
    • Perelman, G.1
  • 40
    • 0036350601 scopus 로고    scopus 로고
    • Dispersive estimates and the 2D cubic NLS equation
    • F. Planchon, Dispersive estimates and the 2D cubic NLS equation, J. Anal. Math. 86 (2002), 319-334.
    • (2002) J. Anal. Math , vol.86 , pp. 319-334
    • Planchon, F.1
  • 41
    • 34948897548 scopus 로고    scopus 로고
    • Existence and stability of the log-log blow-up dynamics for the L2-critical nonlinear Schrödinger equation in a domain
    • F. Planchon and P. Raphaël, Existence and stability of the log-log blow-up dynamics for the L2-critical nonlinear Schrödinger equation in a domain, Ann. Henri Poincaré 8 (2007), no. 6, 1177-1219.
    • (2007) Ann. Henri Poincaré , vol.8 , Issue.6 , pp. 1177-1219
    • Planchon, F.1    Raphaël, P.2
  • 42
    • 13844296411 scopus 로고    scopus 로고
    • Stability of the log-log bound for blow up solutions to the critical non linear Schrödinger equation
    • P. Raphaël, Stability of the log-log bound for blow up solutions to the critical non linear Schrödinger equation, Math. Ann. 331 (2005), no. 3, 577-609.
    • (2005) Math. Ann , vol.331 , Issue.3 , pp. 577-609
    • Raphaël, P.1
  • 43
    • 33748575882 scopus 로고    scopus 로고
    • Existence and stability of a solution blowing up on a sphere for an L2-supercritical nonlinear Schrödinger equation
    • P. Raphaël, Existence and stability of a solution blowing up on a sphere for an L2-supercritical nonlinear Schrödinger equation, Duke Math. J. 134 (2006), no. 2, 199-258.
    • (2006) Duke Math. J , vol.134 , Issue.2 , pp. 199-258
    • Raphaël, P.1
  • 44
    • 84855435413 scopus 로고    scopus 로고
    • Stable blow up dynamics for the critical co-rotational wave maps and equivariant Yang-Mills problems
    • P. Raphaël and I. Rodnianski, Stable blow up dynamics for the critical co-rotational wave maps and equivariant Yang-Mills problems, Inst. Hautes études Sci. Publ. Math. (2012), 1-122.
    • (2012) Inst. Hautes Études Sci. Publ. Math , pp. 1-122
    • Raphaël, P.1    Rodnianski, I.2
  • 45
    • 78651309844 scopus 로고    scopus 로고
    • Existence and uniqueness of minimal blow-up solutions to an inhomogeneous mass critical NLS
    • P. Raphaël and J. Szeftel, Existence and uniqueness of minimal blow-up solutions to an inhomogeneous mass critical NLS, J. Amer. Math. Soc. 24 (2011), no. 2, 471-546.
    • (2011) J. Amer. Math. Soc , vol.24 , Issue.2 , pp. 471-546
    • Raphaël, P.1    Szeftel, J.2
  • 46
    • 70350573248 scopus 로고    scopus 로고
    • Stable manifolds for an orbitally unstable nonlinear Schrödinger equation
    • W. Schlag, Stable manifolds for an orbitally unstable nonlinear Schrödinger equation, Ann. of Math. (2) 169 (2009), no. 1, 139-227.
    • (2009) Ann. of Math , vol.169 , Issue.1-2 , pp. 139-227
    • Schlag, W.1
  • 47
    • 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), no. 3, 705-714.
    • (1977) Duke Math. J , vol.44 , Issue.3 , pp. 705-714
    • Strichartz, R.S.1
  • 48
    • 0003230098 scopus 로고    scopus 로고
    • The Nonlinear Schrödinger Equation. Self-Focusing and Wave Collapse
    • Springer-Verlag, New York
    • C. Sulem and P.-L. Sulem, The Nonlinear Schrödinger Equation. Self-Focusing and Wave Collapse, Appl. Math. Sci., vol. 139, Springer-Verlag, New York, 1999.
    • (1999) Appl. Math. Sc , vol.139
    • Sulem, C.1    Sulem, P.-L.2
  • 49
    • 0041473959 scopus 로고
    • Nonlinear Schrödinger equations and sharp interpolation estimates
    • M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys. 87 (1983), no. 4, 567-576.
    • (1983) Comm. Math. Phys , vol.87 , Issue.4 , pp. 567-576
    • Weinstein, M.I.1
  • 50
    • 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), no. 3, 472-491.
    • (1985) SIAM J. Math. Anal , vol.16 , Issue.3 , pp. 472-491
    • Weinstein, M.I.1
  • 51
    • 0002557939 scopus 로고
    • Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media
    • V. E. Zakharov and A. B. Shabat, Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media, Sov. Phys. JETP 34 (1972), no. 1, 62-69.
    • (1972) Sov. Phys. JETP , vol.34 , Issue.1 , pp. 62-69
    • Zakharov, V.E.1    Shabat, A.B.2


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.