Periodic nonlinear Schrödinger equation and invariant measures

@article{Bourgain1994PeriodicNS,
  title={Periodic nonlinear Schr{\"o}dinger equation and invariant measures},
  author={J. Bourgain},
  journal={Communications in Mathematical Physics},
  year={1994},
  volume={166},
  pages={1-26}
}
  • J. Bourgain
  • Published 1994
  • Mathematics
  • Communications in Mathematical Physics
  • AbstractIn this paper we continue some investigations on the periodic NLSEiuu +iuxx +u|u|p-2 (p≦6) started in [LRS]. We prove that the equation is globally wellposed for a set of data Φ of full normalized Gibbs measrue $$e^{ - \beta H(\phi )} Hd\phi (x),H(\phi ) = \tfrac{1}{2}\int {\left| {\phi '} \right|^2 - \tfrac{1}{p}\int {\left| \phi \right|p} } $$ (after suitableL2-truncation). The set and the measure are invariant under the flow. The proof of a similar result for the KdV and modified… CONTINUE READING
    276 Citations
    Almost Sure Scattering at Mass Regularity for Radial Schr\"odinger Equations
    • 2
    • Highly Influenced
    • PDF
    Invariance of the Gibbs Measure for the Schrödinger-Benjamin-Ono System
    • T. Oh
    • Mathematics, Computer Science
    • SIAM J. Math. Anal.
    • 2009
    • 48
    • PDF
    Probabilistic well-posedness for supercritical wave equation on $\mathbb{T}^3$
    • 5
    • Highly Influenced
    • PDF
    Non-homogeneous Problems for Nonlinear Schr\"odinger Equations in a Strip Domain
    • PDF
    Gibbs Measure Dynamics for the Fractional NLS
    • 7
    • PDF
    Almost sure local well-posedness for a derivative nonlinear wave equation
    • 9
    • PDF

    References

    SHOWING 1-9 OF 9 REFERENCES
    Fourier restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations
    • Geom. and Funct. Anal
    • 1993
    On the invariant measure for the nonlinear Schrόdinger equation
    • Doklady Akad Nauk SSSR 317,
    • 1991
    Invariant measures for the 2 Z ) - defocusing nonlinear Schrόdinger equation
    • Preprint IHES 4 / 94 [ GV ]
    On the longtime behaviour of nonlinear Hamiltonian evolution equations
    • Preprint IHES 4 / 94 , to appear in Geom . and Funct . Anal . ( GAPA )