Lancaster EPrints

Logarithmic Sobolev inequality for the invariant measure of the periodic Korteweg--de Vries equation.

Blower, Gordon (2012) Logarithmic Sobolev inequality for the invariant measure of the periodic Korteweg--de Vries equation. Stochastics: An International Journal of Probability and Stochastic Processes formerly Stochastics and Stochastics Reports, 84 (4). pp. 533-542. ISSN 1744-2508

[img]
Preview
PDF (KdVmeasure.pdf) - Draft Version
Download (132Kb) | Preview

    Abstract

    The periodic KdV equation arises from a Hamiltonian system with infinite-dimensional phase space L^2(T). Bourgain has shown that there exists a Gibbs measure \nu on balls in the phase space such that the Cauchy problem for KdV is well posed on the support of \nu, and \nu is invariant under the KdV flow. This paper shows that \nu satisfies a logarithmic Sobolev inequality. The stationary points of the Hamiltonian on spheres are found in terms of elliptic functions, and they are shown to be linearly stable. The paper also presents logarithmic Sobolev inequalities for the modified period KdV equation and the cubic nonlinear Schrodinger equation for small values of the number operator N.

    Item Type: Article
    Journal or Publication Title: Stochastics: An International Journal of Probability and Stochastic Processes formerly Stochastics and Stochastics Reports
    Additional Information: The final, definitive version of this article has been published in the Journal, Stochastics: An International Journal of Probability and Stochastic Processes, 84 (4), 2012, Informa Plc
    Uncontrolled Keywords: Gibbs measure ; concentration inequality ; nonlinear Schrodinger equation
    Subjects: Q Science > QA Mathematics
    Departments: Faculty of Science and Technology > Mathematics and Statistics
    ID Code: 31115
    Deposited By: Professor Gordon Blower
    Deposited On: 21 Dec 2009 09:22
    Refereed?: Yes
    Published?: Published
    Last Modified: 08 Dec 2014 10:30
    Identification Number:
    URI: http://eprints.lancs.ac.uk/id/eprint/31115

    Actions (login required)

    View Item