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A solution to the focusing 3d NLS that blows up on a contracting sphere


Authors: Justin Holmer, Galina Perelman and Svetlana Roudenko
Journal: Trans. Amer. Math. Soc. 367 (2015), 3847-3872
MSC (2010): Primary 35Q55
DOI: https://doi.org/10.1090/S0002-9947-2015-06057-7
Published electronically: February 20, 2015
MathSciNet review: 3324912
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Abstract: We rigorously construct radial $ H^1$ solutions to the 3d cubic focusing NLS equation $ i\partial _t \psi + \Delta \psi + 2 \vert\psi \vert^2\psi =0$ that blow-up along a contracting sphere. With blow-up time set to $ t=0$, the solutions concentrate on a sphere at radius $ \sim t^{1/3}$ but focus towards this sphere at the faster rate $ \sim t^{2/3}$. Such dynamics were originally proposed heuristically by Degtyarev-Zakharov-Rudakov in 1975 and independently later by Holmer-Roudenko in 2007, where it was demonstrated to be consistent with all conservation laws of this equation. In the latter paper, it was proposed as a solution that would yield divergence of the $ L_x^3$ norm within the ``wide'' radius $ \sim \Vert\nabla u(t)\Vert _{L_x^2}^{-1/2}$ but not within the ``tight'' radius $ \sim \Vert\nabla u(t)\Vert _{L_x^2}^{-2}$, the second being the rate of contraction of self-similar blow-up solutions observed numerically and described in detail by Sulem-Sulem in 1999.


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  • [1] Jean 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 (1998). Dedicated to Ennio De Giorgi. MR 1655515 (99m:35219)
  • [2] L.M. Degtyarev, V.E. Zakharov, and L.V. Rudakov, Two examples of Langmuir wave collapse, Sov. Phys. JETP 41 (1975), pp. 57-61.
  • [3] Gadi Fibich, Nir Gavish, and Xiao-Ping Wang, Singular ring solutions of critical and supercritical nonlinear Schrödinger equations, Phys. D 231 (2007), no. 1, 55-86. MR 2370365 (2008k:35440), https://doi.org/10.1016/j.physd.2007.04.007
  • [4] J. Holmer and Q.-H. Lin, Phase-driven interaction of widely separated nonlinear Schrödinger solitons, to appear in Journal of Hyperbolic Differential Equations, available at arXiv:1108.4859 [math.AP].
  • [5] Justin Holmer and Svetlana Roudenko, On blow-up solutions to the 3D cubic nonlinear Schrödinger equation, Appl. Math. Res. Express. AMRX 1 (2007), Art. ID abm004, 31. MR 2354447 (2008i:35227)
  • [6] Justin Holmer, Rodrigo Platte, and Svetlana Roudenko, Blow-up criteria for the 3D cubic nonlinear Schrödinger equation, Nonlinearity 23 (2010), no. 4, 977-1030. MR 2630088 (2011c:35538), https://doi.org/10.1088/0951-7715/23/4/011
  • [7] Justin Holmer and Maciej Zworski, Slow soliton interaction with delta impurities, J. Mod. Dyn. 1 (2007), no. 4, 689-718. MR 2342704 (2008k:35446), https://doi.org/10.3934/jmd.2007.1.689
  • [8] F. Merle, P. Raphaël, and J. Szeftel, On collapsing ring blow up solutions to the mass supercritical NLS, arXiv:1202.5218 [math.AP].
  • [9] Galina Perelman, Analysis seminar, Université de Cergy-Pontoise, Dec 2011 (joint work with J. Holmer and S. Roudenko).
  • [10] Walter A. Strauss, Nonlinear wave equations, CBMS Regional Conference Series in Mathematics, vol. 73, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1989. MR 1032250 (91g:35002)
  • [11] Terence Tao, Nonlinear dispersive equations, CBMS Regional Conference Series in Mathematics, vol. 106, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 2006. Local and global analysis. MR 2233925 (2008i:35211)
  • [12] Michael I. Weinstein, Lyapunov stability of ground states of nonlinear dispersive evolution equations, Comm. Pure Appl. Math. 39 (1986), no. 1, 51-67. MR 820338 (87f:35023), https://doi.org/10.1002/cpa.3160390103
  • [13] Catherine Sulem and Pierre-Louis Sulem, The nonlinear Schrödinger equation, Applied Mathematical Sciences, vol. 139, Springer-Verlag, New York, 1999. Self-focusing and wave collapse. MR 1696311 (2000f:35139)

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Additional Information

Justin Holmer
Affiliation: Department of Mathematics, Brown University, Box 1917, 151 Thayer Street, Providence, Rhode Island 02912
Email: holmer@math.brown.edu

Galina Perelman
Affiliation: Laboratoire d’Analyse et de Mathématiques Appliquées, Université Paris-Est Créteil, Créteil Cedex, France
Email: galina.perelman@u-pec.fr

Svetlana Roudenko
Affiliation: Department of Mathematics, Munroe Hall, The George Washington University, 2115 G Street NW, Washington, DC 20052
Email: roudenko@gwu.edu

DOI: https://doi.org/10.1090/S0002-9947-2015-06057-7
Received by editor(s): October 4, 2012
Received by editor(s) in revised form: December 14, 2012
Published electronically: February 20, 2015
Article copyright: © Copyright 2015 American Mathematical Society

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