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Volumn 4, Issue 5, 2011, Pages 677-727

Standing ring blowup solutions for cubic nonlinear schrödinger equations

Author keywords

Blowup rate; Blowup set; Codimension; Collapse; Focusing; Log log rate; Nonlinear Schr dinger equation; Regularity; Supercritical

Indexed keywords


EID: 84878693522     PISSN: 21575045     EISSN: 1948206X     Source Type: Journal    
DOI: 10.2140/apde.2011.4.677     Document Type: Article
Times cited : (8)

References (38)
  • 1
    • 0020591567 scopus 로고
    • Nonlinear scalar field equations. I. Existence of a ground state
    • MR 84h:35054a
    • H. Berestycki and P.-L. Lions, "Nonlinear scalar field equations. I. Existence of a ground state", Arch. Rational Mech. Anal. 82:4 (1983), 313-345. MR 84h:35054a
    • (1983) Arch. Rational Mech. Anal. , vol.82 , Issue.4 , pp. 313-345
    • Berestycki, H.1    Lions, P.-L.2
  • 2
    • 84977279911 scopus 로고    scopus 로고
    • Semilinear Schrödinger equations, Courant Lecture Notes in Mathematics 10
    • CIMS, New York, MR 2004j:35266 Zbl 1055.35003
    • T. Cazenave, Semilinear Schrödinger equations, Courant Lecture Notes in Mathematics 10, CIMS, New York, 2003. MR 2004j:35266 Zbl 1055.35003
    • (2003) , vol.10
    • Cazenave, T.1
  • 3
    • 0015249887 scopus 로고
    • 3=0 and a variational characterization of other solutions
    • MR 48 #11814 Zbl 0249.35029
    • 3=0 and a variational characterization of other solutions", Arch. Rational Mech. Anal. 46 (1972), 81-95. MR 48 #11814 Zbl 0249.35029
    • (1972) Arch. Rational Mech. Anal. , vol.46 , pp. 81-95
    • Coffman, C.V.1
  • 4
    • 63949086654 scopus 로고    scopus 로고
    • Dynamic of threshold solutions for energy-critical NLS
    • MR 2010e:35258 Zbl 05603849
    • T. Duyckaerts and F. Merle, "Dynamic of threshold solutions for energy-critical NLS", Geom. Funct. Anal. 18:6 (2009), 1787-1840. MR 2010e:35258 Zbl 05603849
    • (2009) Geom. Funct. Anal. , vol.18 , Issue.6 , pp. 1787-1840
    • Duyckaerts, T.1    Merle, F.2
  • 5
    • 77951143619 scopus 로고    scopus 로고
    • Threshold solutions for the focusing 3D cubic Schrödinger equation
    • MR 2011c:35533 Zbl 1195.35276
    • T. Duyckaerts and S. Roudenko, "Threshold solutions for the focusing 3D cubic Schrödinger equation", Rev. Mat. Iberoam. 26:1 (2010), 1-56. MR 2011c:35533 Zbl 1195.35276
    • (2010) Rev. Mat. Iberoam. , vol.26 , Issue.1 , pp. 1-56
    • Duyckaerts, T.1    Roudenko, S.2
  • 6
    • 59349111935 scopus 로고    scopus 로고
    • Scattering for the non-radial 3D cubic nonlinear Schrödinger equation
    • MR 2010e:35257
    • T. Duyckaerts, J. Holmer, and S. Roudenko, "Scattering for the non-radial 3D cubic nonlinear Schrödinger equation", Math. Res. Lett. 15:6 (2008), 1233-1250. MR 2010e:35257
    • (2008) Math. Res. Lett. , vol.15 , Issue.6 , pp. 1233-1250
    • Duyckaerts, T.1    Holmer, J.2    Roudenko, S.3
  • 7
    • 33746745308 scopus 로고    scopus 로고
    • Proof of a spectral property related to the singularity formation for the L2 critical nonlinear Schrödinger equation
    • MR 2007d:35254 Zbl 1100.35097
    • G. Fibich, F. Merle, and P. Raphaël, "Proof of a spectral property related to the singularity formation for the L2 critical nonlinear Schrödinger equation", Phys. D 220:1 (2006), 1-13. MR 2007d:35254 Zbl 1100.35097
    • (2006) Phys. D , vol.220 , Issue.1 , pp. 1-13
    • Fibich, G.1    Merle, F.2    Raphaël, P.3
  • 8
    • 34249993538 scopus 로고    scopus 로고
    • Singular ring solutions of critical and supercritical nonlinear Schrödinger equations
    • MR 2008k:35440 Zbl 1118.35043
    • G. Fibich, N. Gavish, and X.-P. Wang, "Singular ring solutions of critical and supercritical nonlinear Schrödinger equations", Phys. D 231:1 (2007), 55-86. MR 2008k:35440 Zbl 1118.35043
    • (2007) Phys. D , vol.231 , Issue.1 , pp. 55-86
    • Fibich, G.1    Gavish, N.2    Wang, X.-P.3
  • 9
    • 77955700830 scopus 로고    scopus 로고
    • On blow-up solutions to the 3D cubic nonlinear Schrödinger equation
    • ID abm004. MR 2008i:35227
    • J. Holmer and S. Roudenko, "On blow-up solutions to the 3D cubic nonlinear Schrödinger equation", Appl. Math. Res. Express. AMRX 2007:1 (2007), Art. ID abm004. MR 2008i:35227
    • (2007) Appl. Math. Res. Express. AMRX , vol.2007 , Issue.1
    • Holmer, J.1    Roudenko, S.2
  • 10
    • 47749109045 scopus 로고    scopus 로고
    • A sharp condition for scattering of the radial 3D cubic nonlinear Schrödinger equation
    • MR 2009h:35403 Zbl 1155.35094
    • J. Holmer and S. Roudenko, "A sharp condition for scattering of the radial 3D cubic nonlinear Schrödinger equation", Comm. Math. Phys. 282:2 (2008), 435-467. MR 2009h:35403 Zbl 1155.35094
    • (2008) Comm. Math. Phys. , vol.282 , Issue.2 , pp. 435-467
    • Holmer, J.1    Roudenko, S.2
  • 11
    • 77950937063 scopus 로고    scopus 로고
    • Divergence of infinite-variance nonradial solutions to the 3D NLS equation
    • MR 2011m:35353 Zbl 1195.35277
    • J. Holmer and S. Roudenko, "Divergence of infinite-variance nonradial solutions to the 3D NLS equation", Comm. Partial Differential Equations 35:5 (2010), 878-905. MR 2011m:35353 Zbl 1195.35277
    • (2010) Comm. Partial Differential Equations , vol.35 , Issue.5 , pp. 878-905
    • Holmer, J.1    Roudenko, S.2
  • 12
    • 79952911434 scopus 로고    scopus 로고
    • A class of solutions to the 3D cubic nonlinear Schrödinger equation that blows up on a circle
    • MR 2012b:35323 Zbl 1220.35159
    • J. Holmer and S. Roudenko, "A class of solutions to the 3D cubic nonlinear Schrödinger equation that blows up on a circle", Appl. Math. Res. Express. AMRX 2011:1 (2011), 23-94. MR 2012b:35323 Zbl 1220.35159
    • (2011) Appl. Math. Res. Express. AMRX , vol.2011 , Issue.1 , pp. 23-94
    • Holmer, J.1    Roudenko, S.2
  • 13
    • 77949739721 scopus 로고    scopus 로고
    • Blow-up criteria for the 3D cubic nonlinear Schrödinger equation
    • MR 2011c:35538 Zbl 1202.37105
    • J. Holmer, R. Platte, and S. Roudenko, "Blow-up criteria for the 3D cubic nonlinear Schrödinger equation", Nonlinearity 23:4 (2010), 977-1030. MR 2011c:35538 Zbl 1202.37105
    • (2010) Nonlinearity , vol.23 , Issue.4 , pp. 977-1030
    • Holmer, J.1    Platte, R.2    Roudenko, S.3
  • 14
    • 33750526878 scopus 로고    scopus 로고
    • Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case
    • MR 2007g:35232
    • 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. 166:3 (2006), 645-675. MR 2007g:35232
    • (2006) Invent. Math. , vol.166 , Issue.3 , pp. 645-675
    • Kenig, C.E.1    Merle, F.2
  • 15
    • 74549167807 scopus 로고    scopus 로고
    • The cubic nonlinear Schrödinger equation in two dimensions with radial data
    • MR 2010m:35487 Zbl 1187.35237
    • R. Killip, T. Tao, and M. Visan, "The cubic nonlinear Schrödinger equation in two dimensions with radial data", J. Eur. Math. Soc. .JEMS/ 11:6 (2009), 1203-1258. MR 2010m:35487 Zbl 1187.35237
    • (2009) J. Eur. Math. Soc. .JEMS/ , vol.11 , Issue.6 , pp. 1203-1258
    • Killip, R.1    Tao, T.2    Visan, M.3
  • 16
    • 34249974409 scopus 로고
    • Uniqueness of positive solutions of δu-uCup D0 in Rn
    • MR 90d:35015 Zbl 0676.35032
    • M. K. Kwong, "Uniqueness of positive solutions of δu-uCup D0 in Rn", Arch. Rational Mech. Anal. 105:3 (1989), 243-266. MR 90d:35015 Zbl 0676.35032
    • (1989) Arch. Rational Mech. Anal. , vol.105 , Issue.3 , pp. 243-266
    • Kwong, M.K.1
  • 17
    • 0000893351 scopus 로고
    • Rate of blowup for solutions of the nonlinear Schrödinger equation at critical dimension
    • MR 89k:35218
    • 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/ 38:8 (1988), 3837-3843. MR 89k:35218
    • (1988) Phys. Rev. A .3/ , vol.38 , Issue.8 , pp. 3837-3843
    • Landman, M.J.1    Papanicolaou, G.C.2    Sulem, C.3    Sulem, P.-L.4
  • 18
    • 0000296501 scopus 로고
    • Symétrie et compacité dans les espaces de Sobolev
    • MR 84k:46027 Zbl 0501.46032
    • P.-L. Lions, "Symétrie et compacité dans les espaces de Sobolev", J. Funct. Anal. 49:3 (1982), 315-334. MR 84k:46027 Zbl 0501.46032
    • (1982) J. Funct. Anal. , vol.49 , Issue.3 , pp. 315-334
    • Lions, P.-L.1
  • 19
    • 0036922015 scopus 로고    scopus 로고
    • Existence of nonstationary bubbles in higher dimensions
    • MR 2004e:35213 Zbl 1040.35116
    • M. Mari̧s, "Existence of nonstationary bubbles in higher dimensions", J. Math. Pures Appl. .9/ 81:12 (2002), 1207-1239. MR 2004e:35213 Zbl 1040.35116
    • (2002) J. Math. Pures Appl. .9/ , vol.81 , Issue.12 , pp. 1207-1239
    • Mari̧s, M.1
  • 20
    • 0035635809 scopus 로고    scopus 로고
    • Instability of solitons for the critical generalized Korteweg-de Vries equation
    • MR 2002g:35182 Zbl 0985.35071
    • Y. Martel and F. Merle, "Instability of solitons for the critical generalized Korteweg-de Vries equation", Geom. Funct. Anal. 11:1 (2001), 74-123. MR 2002g:35182 Zbl 0985.35071
    • (2001) Geom. Funct. Anal. , vol.11 , Issue.1 , pp. 74-123
    • Martel, Y.1    Merle, F.2
  • 21
    • 0000041774 scopus 로고
    • Uniqueness of positive radial solutions of δuCf .u/ D 0 in Rn. II
    • MR 94b:35105 Zbl 0804.35034
    • K. McLeod, "Uniqueness of positive radial solutions of δuCf .u/ D 0 in Rn. II", Trans. Amer. Math. Soc. 339:2 (1993), 495-505. MR 94b:35105 Zbl 0804.35034
    • (1993) Trans. Amer. Math. Soc. , vol.339 , Issue.2 , pp. 495-505
    • McLeod, K.1
  • 22
    • 84974001368 scopus 로고
    • Determination of blow-up solutions with minimal mass for nonlinear Schrödinger equations with critical power
    • MR 94b:35262 Zbl 0808.35141
    • F. Merle, "Determination of blow-up solutions with minimal mass for nonlinear Schrödinger equations with critical power", Duke Math. J. 69:2 (1993), 427-454. MR 94b:35262 Zbl 0808.35141
    • (1993) Duke Math. J. , vol.69 , Issue.2 , pp. 427-454
    • Merle, F.1
  • 23
    • 0041350485 scopus 로고    scopus 로고
    • Sharp upper bound on the blow-up rate for the critical nonlinear Schrödinger equation
    • MR 2005j:35207 Zbl 1061.35135
    • F. Merle and P. Raphael, "Sharp upper bound on the blow-up rate for the critical nonlinear Schrödinger equation", Geom. Funct. Anal. 13:3 (2003), 591-642. MR 2005j:35207 Zbl 1061.35135
    • (2003) Geom. Funct. Anal. , vol.13 , Issue.3 , pp. 591-642
    • Merle, F.1    Raphael, P.2
  • 24
    • 2942586742 scopus 로고    scopus 로고
    • On universality of blow-up profile for L2 critical nonlinear Schrödinger equation
    • MR 2006a:35283 Zbl 1067.35110
    • F. Merle and P. Raphael, "On universality of blow-up profile for L2 critical nonlinear Schrödinger equation", Invent. Math. 156:3 (2004), 565-672. MR 2006a:35283 Zbl 1067.35110
    • (2004) Invent. Math. , vol.156 , Issue.3 , pp. 565-672
    • Merle, F.1    Raphael, P.2
  • 25
    • 20444437961 scopus 로고    scopus 로고
    • The blow-up dynamic and upper bound on the blow-up rate for critical nonlinear Schrödinger equation
    • MR 2006k:35277 Zbl 1185.35263
    • 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/ 161:1 (2005), 157-222. MR 2006k:35277 Zbl 1185.35263
    • (2005) Ann. of Math. .2/ , vol.161 , Issue.1 , pp. 157-222
    • Merle, F.1    Raphael, P.2
  • 26
    • 12444281836 scopus 로고    scopus 로고
    • Profiles and quantization of the blow up mass for critical nonlinear Schrödinger equation
    • MR 2006m:35346 Zbl 1062.35137
    • F. Merle and P. Raphael, "Profiles and quantization of the blow up mass for critical nonlinear Schrödinger equation", Comm. Math. Phys. 253:3 (2005), 675-704. MR 2006m:35346 Zbl 1062.35137
    • (2005) Comm. Math. Phys. , vol.253 , Issue.3 , pp. 675-704
    • Merle, F.1    Raphael, P.2
  • 27
    • 30644464564 scopus 로고    scopus 로고
    • On a sharp lower bound on the blow-up rate for the L2 critical nonlinear Schrödinger equation
    • MR 2006j:35223 Zbl 1075.35077
    • F. Merle and P. Raphael, "On a sharp lower bound on the blow-up rate for the L2 critical nonlinear Schrödinger equation", J. Amer. Math. Soc. 19:1 (2006), 37-90. MR 2006j:35223 Zbl 1075.35077
    • (2006) J. Amer. Math. Soc. , vol.19 , Issue.1 , pp. 37-90
    • Merle, F.1    Raphael, P.2
  • 28
    • 80051968524 scopus 로고    scopus 로고
    • Global dynamics above the ground state energy for the cubic NLS equation in 3D
    • preprint, arXiv 1007.4025
    • K. Nakanishi and W. Schlag, "Global dynamics above the ground state energy for the cubic NLS equation in 3D", preprint, 2010. arXiv 1007.4025
    • (2010)
    • Nakanishi, K.1    Schlag, W.2
  • 29
    • 0000835523 scopus 로고
    • Blow-up of H1 solution for the nonlinear Schrödinger equation
    • MR 92k:35262 Zbl 0739.35093
    • T. Ogawa and Y. Tsutsumi, "Blow-up of H1 solution for the nonlinear Schrödinger equation", J. Differential Equations 92:2 (1991), 317-330. MR 92k:35262 Zbl 0739.35093
    • (1991) J. Differential Equations , vol.92 , Issue.2 , pp. 317-330
    • Ogawa, T.1    Tsutsumi, Y.2
  • 30
    • 0035537074 scopus 로고    scopus 로고
    • On the formation of singularities in solutions of the critical nonlinear Schrödinger equation
    • MR 2002m:35205 Zbl 1007.35087
    • G. Perelman, "On the formation of singularities in solutions of the critical nonlinear Schrödinger equation", Ann. Henri Poincaré 2:4 (2001), 605-673. MR 2002m:35205 Zbl 1007.35087
    • (2001) Ann. Henri Poincaré , vol.2 , Issue.4 , pp. 605-673
    • Perelman, G.1
  • 31
    • 0000452161 scopus 로고
    • Compact linear mappings between interpolation spaces
    • MR 29 #3871 Zbl 0128.35204
    • A. Persson, "Compact linear mappings between interpolation spaces", Ark. Mat. 5 (1964), 215-219. MR 29 #3871 Zbl 0128.35204
    • (1964) Ark. Mat. , vol.5 , pp. 215-219
    • Persson, A.1
  • 32
    • 13844296411 scopus 로고    scopus 로고
    • Stability of the log-log bound for blow up solutions to the critical non linear Schrödinger equation
    • MR 2006b:35303
    • P. Raphael, "Stability of the log-log bound for blow up solutions to the critical non linear Schrödinger equation", Math. Ann. 331:3 (2005), 577-609. MR 2006b:35303
    • (2005) Math. Ann. , vol.331 , Issue.3 , pp. 577-609
    • Raphael, P.1
  • 33
    • 33748575882 scopus 로고    scopus 로고
    • Existence and stability of a solution blowing up on a sphere for an L2-supercritical nonlinear Schrödinger equation
    • MR 2007k:35472 Zbl 1117.35077
    • P. Raphaël, "Existence and stability of a solution blowing up on a sphere for an L2-supercritical nonlinear Schrödinger equation", Duke Math. J. 134:2 (2006), 199-258. MR 2007k:35472 Zbl 1117.35077
    • (2006) Duke Math. J. , vol.134 , Issue.2 , pp. 199-258
    • Raphaël, P.1
  • 34
    • 70350025320 scopus 로고    scopus 로고
    • Standing ring blow up solutions to the N-dimensional quintic nonlinear Schrödinger equation
    • MR 2010k:35469 Zbl 1184.35295
    • P. Raphaël and J. Szeftel, "Standing ring blow up solutions to the N-dimensional quintic nonlinear Schrödinger equation", Comm. Math. Phys. 290:3 (2009), 973-996. MR 2010k:35469 Zbl 1184.35295
    • (2009) Comm. Math. Phys. , vol.290 , Issue.3 , pp. 973-996
    • Raphaël, P.1    Szeftel, J.2
  • 35
    • 0003230098 scopus 로고    scopus 로고
    • The nonlinear Schrödinger equation: self-focusing and wave collapse
    • Springer, New York,MR 2000f:35139 Zbl 0928.35157
    • C. Sulem and P.-L. Sulem, The nonlinear Schrödinger equation: self-focusing and wave collapse, Applied Mathematical Sciences 139, Springer, New York, 1999. MR 2000f:35139 Zbl 0928.35157
    • (1999) Applied Mathematical Sciences , vol.139
    • Sulem, C.1    Sulem, P.-L.2
  • 36
    • 69749083744 scopus 로고    scopus 로고
    • Nonlinear dispersive equations, CBMS Regional Conference Series in Mathematics: local and global analysis 106
    • Providence, RI, MR 2008i:35211 Zbl 1106.35001
    • T. Tao, Nonlinear dispersive equations, CBMS Regional Conference Series in Mathematics: local and global analysis 106, Amer. Math. Soc., Providence, RI, 2006. MR 2008i:35211 Zbl 1106.35001
    • (2006) Amer. Math. Soc.
    • Tao, T.1
  • 37
    • 0041473959 scopus 로고
    • Nonlinear Schrödinger equations and sharp interpolation estimates
    • MR 84d:35140 Zbl 0527.35023
    • M. I. Weinstein, "Nonlinear Schrödinger equations and sharp interpolation estimates", Comm. Math. Phys. 87:4 (1983), 567-576. MR 84d:35140 Zbl 0527.35023
    • (1983) Comm. Math. Phys. , vol.87 , Issue.4 , pp. 567-576
    • Weinstein, M.I.1
  • 38
    • 0000686130 scopus 로고
    • Modulational stability of ground states of nonlinear Schrödinger equations
    • MR 86i:35130 Zbl 0583.35028
    • M. I.Weinstein, "Modulational stability of ground states of nonlinear Schrödinger equations", SIAM J. Math. Anal. 16:3 (1985), 472-491. MR 86i:35130 Zbl 0583.35028
    • (1985) SIAM J. Math. Anal. , vol.16 , Issue.3 , pp. 472-491
    • Weinstein, M.I.1


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