Analysis & PDE

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

Two-dimensional nonlinear Schrödinger equation with random radial data

Yu Deng

Full-text: Open access

Abstract

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.apde/1513731261

Digital Object Identifier
doi:10.2140/apde.2012.5.913

Mathematical Reviews number (MathSciNet)
MR3022846

Zentralblatt MATH identifier
1264.35212

Subjects
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

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

Citation

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. https://projecteuclid.org/euclid.apde/1513731261


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