Analysis & PDE

  • Anal. PDE
  • Volume 5, Number 5 (2012), 913-960.

Two-dimensional nonlinear Schrödinger equation with random radial data

Yu Deng

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We study radial solutions of a certain two-dimensional nonlinear Schrödinger (NLS) equation with harmonic potential, which is supercritical with respect to the initial data. By combining the nonlinear smoothing effect of Schrödinger equation with Lp estimates of Laguerre functions, we are able to prove an almost-sure global well-posedness result and the invariance of the Gibbs measure. We also discuss an application to the NLS equation without harmonic potential.

Article information

Anal. PDE, Volume 5, Number 5 (2012), 913-960.

Received: 16 November 2010
Revised: 14 February 2011
Accepted: 3 June 2011
First available in Project Euclid: 20 December 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35Q55: NLS-like equations (nonlinear Schrödinger) [See also 37K10] 37L40: Invariant measures 37L50: Noncompact semigroups; dispersive equations; perturbations of Hamiltonian systems
Secondary: 37K05: Hamiltonian structures, symmetries, variational principles, conservation laws

nonlinear Schrödinger equation supercritical NLS random data Gibbs measure global well-posedness


Deng, Yu. Two-dimensional nonlinear Schrödinger equation with random radial data. Anal. PDE 5 (2012), no. 5, 913--960. doi:10.2140/apde.2012.5.913.

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