Duke Mathematical Journal

Existence and stability of a solution blowing up on a sphere for an L2-supercritical nonlinear Schrödinger equation

Pierre Raphaël

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We consider the quintic two-dimensional focusing nonlinear Schrödinger equation iut=Δu|u|4u which is L2-supercritical. Even though the existence of finite-time blow-up solutions in the energy space H1 is known, very little is understood concerning the singularity formation. Numerics suggest the existence of a stable blow-up dynamic corresponding to a self-similar blowup at one point in space. We prove the existence of a different type of dynamic and exhibit an open set among the H1-radial distributions of initial data for which the corresponding solution blows up in finite time on a sphere. This is the first result of an explicit description of a blow-up dynamic in the L2-supercritical setting

Article information

Source
Duke Math. J., Volume 134, Number 2 (2006), 199-258.

Dates
First available in Project Euclid: 8 August 2006

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1155045502

Digital Object Identifier
doi:10.1215/S0012-7094-06-13421-X

Mathematical Reviews number (MathSciNet)
MR2248831

Zentralblatt MATH identifier
1117.35077

Subjects
Primary: 35Q55: NLS-like equations (nonlinear Schrödinger) [See also 37K10]
Secondary: 35Q51: Soliton-like equations [See also 37K40] 35B05: Oscillation, zeros of solutions, mean value theorems, etc.

Citation

Raphaël, Pierre. Existence and stability of a solution blowing up on a sphere for an $L^2$ -supercritical nonlinear Schrödinger equation. Duke Math. J. 134 (2006), no. 2, 199--258. doi:10.1215/S0012-7094-06-13421-X. https://projecteuclid.org/euclid.dmj/1155045502


Export citation

References

  • H. Berestycki and P.-L. Lions, Nonlinear scalar field equations, I: Existence of a ground state, Arch. Rational Mech. Anal. 82 (1983), 313--345.
  • J. Bergh and J. LöFströM, Interpolation Spaces: An Introduction, Grundlehren Math. Wiss. 223, Springer, Berlin, 1976.
  • J. Bourgain, ``Problems in Hamiltonian PDE's'' in GAFA 2000 (Tel Aviv, 1999), Geom. Funct. Anal. 2000, special volume, part 1, Birkhäuser, Basel, 32--56.
  • 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), 197--215.
  • T. Cazenave, Semilinear Schrödinger Equations, Courant Lect. Notes Math. 10, Courant Inst. Math. Sci., New York Univ., New York, 2003.
  • T. Cazenave and P.-L. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations, Comm. Math. Phys. 85 (1982), 549--561.
  • T. Cazenave and F. B. Weissler, ``Some remarks on the nonlinear Schrödinger equation in the critical case'' in Nonlinear Semigroups, Partial Differential Equations and Attractors (Washington, D.C., 1987), Lecture Notes in Math. 1394, Springer, Berlin, 1989, 18--29.
  • M. Christ and A. Kiselev, Maximal functions associated to filtrations, J. Funct. Anal. 179 (2001), 409--425.
  • G. Fibich, F. Merle, and P. RaphaëL, Numerical proof of a spectral property related to the singularity formation for the $L^2$ critical nonlinear Schrödinger equation, to appear in Phys. D., preprint, 2005.
  • Y. Giga and R. V. Kohn, Nondegeneracy of blowup for semilinear heat equations, Comm. Pure Appl. Math. 42 (1989), 845--884.
  • J. Ginibre and G. Velo, On a class of nonlinear Schrödinger equations, I: The Cauchy problem, general case, J. Funct. Anal. 32 (1979), 1--32.
  • —, The global Cauchy problem for the nonlinear Schrödinger equation revisited, Ann. Inst. H. Poincaré Anal. Non Linéaire 2 (1985), 309--327.
  • R. T. Glassey, On the blowing up of solutions to the Cauchy problem for nonlinear Schrödinger equations, J. Math. Phys. 18 (1977), 1794--1797.
  • T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math. 41 (1988), 891--907.
  • M. K. Kwong, Uniqueness of positive solutions of $\Delta u-u+u\sp p=0$ in $R\sp n$, Arch. Rational Mech. Anal. 105 (1989), 243--266.
  • 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), 3837--3843.
  • P.-L. Lions, Symétrie et compacité dans les espaces de Sobolev, J. Funct. Anal. 49 (1982), 315--334.
  • F. Merle, Determination of blow-up solutions with minimal mass for nonlinear Schrödinger equations with critical power, Duke Math. J. 69 (1993), 427--454.
  • F. Merle and P. RaphaëL, Sharp upper bound on the blow-up rate for the critical nonlinear Schrödinger equation, Geom. Funct. Anal. 13 (2003), 591--642.
  • —, On universality of blow-up profile for $L^2$ critical nonlinear Schrödinger equation, Invent. Math. 156 (2004), 565--672.
  • —, Blow-up dynamic and upper bound on the blow-up rate for critical nonlinear Schrödinger equation, Ann. of Math. (2) 161 (2005), 157--222.
  • —, Profiles and quantization of the blow up mass for critical nonlinear Schrödinger equation, Comm. Math. Phys. 253 (2005), 675--704.
  • —, On a sharp lower bound on the blow-up rate for the $L^2$ critical nonlinear Schrödinger equation, J. Amer. Math. Soc. 19 (2006), 37--90.
  • G. Perelman, On the formation of singularities in solutions of the critical nonlinear Schrödinger equation, Ann. Henri Poincaré 2 (2001), 605--673.
  • P. RaphaëL, Stability of the log-log bound for blow up solutions to the critical non linear Schrödinger equation, Math. Ann. 331 (2005), 577--609.
  • R. S. Strichartz, Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke Math. J. 44 (1977), 705--714.
  • C. Sulem and P.-L. Sulem, The Nonlinear Schrödinger Equation: Self-Focusing and Wave Collapse, Appl. Math. Sci. 139, Springer, New York, 1999.
  • M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys. 87 (1982/83), 567--576.
  • —, Modulational stability of ground states of nonlinear Schrödinger equations, SIAM J. Math. Anal. 16 (1985), 472--491.
  • V. E. Zakharov and A. B. Shabat, Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media, Soviet Physics JETP 34 (1972), 62--69.