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

Research output: Contribution to journalJournal article

Published

Journal publication date2012
JournalStochastics: An International Journal of Probability and Stochastic Processes formerly Stochastics and Stochastics Reports
Journal number4
Volume84
Number of pages10
Pages533-542
Early online date29/09/11
Original languageEnglish

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.

Bibliographic note

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