A solution to the focusing 3d NLS that blows up on a contracting sphere
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- by Justin Holmer, Galina Perelman and Svetlana Roudenko PDF
- Trans. Amer. Math. Soc. 367 (2015), 3847-3872 Request permission
Abstract:
We rigorously construct radial $H^1$ solutions to the 3d cubic focusing NLS equation $i\partial _t \psi + \Delta \psi + 2 |\psi |^2\psi =0$ 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 $\sim \|\nabla u(t)\|_{L_x^2}^{-1/2}$ but not within the “tight” radius $\sim \|\nabla u(t)\|_{L_x^2}^{-2}$, the second being the rate of contraction of self-similar blow-up solutions observed numerically and described in detail by Sulem-Sulem in 1999.References
- 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
- L.M. Degtyarev, V.E. Zakharov, and L.V. Rudakov, Two examples of Langmuir wave collapse, Sov. Phys. JETP 41 (1975), pp. 57–61.
- 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, DOI 10.1016/j.physd.2007.04.007
- 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].
- Justin Holmer and Svetlana Roudenko, On blow-up solutions to the 3D cubic nonlinear Schrödinger equation, Appl. Math. Res. Express. AMRX (2007), Art. ID abm004, 31. [Issue information previously given as no. 1 (2007)]. MR 2354447
- 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, DOI 10.1088/0951-7715/23/4/011
- Justin Holmer and Maciej Zworski, Slow soliton interaction with delta impurities, J. Mod. Dyn. 1 (2007), no. 4, 689–718. MR 2342704, DOI 10.3934/jmd.2007.1.689
- F. Merle, P. Raphaël, and J. Szeftel, On collapsing ring blow up solutions to the mass supercritical NLS, arXiv:1202.5218 [math.AP].
- Galina Perelman, Analysis seminar, Université de Cergy-Pontoise, Dec 2011 (joint work with J. Holmer and S. Roudenko).
- 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; by the American Mathematical Society, Providence, RI, 1989. MR 1032250
- Terence Tao, Nonlinear dispersive equations, CBMS Regional Conference Series in Mathematics, vol. 106, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2006. Local and global analysis. MR 2233925, DOI 10.1090/cbms/106
- 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, DOI 10.1002/cpa.3160390103
- 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
Additional Information
- Justin Holmer
- Affiliation: Department of Mathematics, Brown University, Box 1917, 151 Thayer Street, Providence, Rhode Island 02912
- MR Author ID: 759238
- 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
- MR Author ID: 701923
- Email: roudenko@gwu.edu
- Received by editor(s): October 4, 2012
- Received by editor(s) in revised form: December 14, 2012
- Published electronically: February 20, 2015
- © Copyright 2015 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 367 (2015), 3847-3872
- MSC (2010): Primary 35Q55
- DOI: https://doi.org/10.1090/S0002-9947-2015-06057-7
- MathSciNet review: 3324912