Dynamics of Partial Differential Equations

Volume 13 (2016)

Number 2

On invariant Gibbs measures for the generalized KdV equations

Pages: 133 – 153

DOI: http://dx.doi.org/10.4310/DPDE.2016.v13.n2.a3

Authors

Tadahiro Oh (School of Mathematics, University of Edinburgh, and Maxwell Institute for the Mathematical Sciences, Edinburgh, Scotland)

Geordie Richards (Department of Mathematics, University of Rochester, New York, U.S.A.)

Laurent Thomann (Institut Élie Cartan, Université de Lorraine, Vandoeuvre-lès-Nancy, France)

Abstract

We consider the defocusing generalized KdV equations on the circle. In particular, we construct global-in-time solutions with initial data distributed according to the Gibbs measure and show that the law of the random solutions, at any time, is again given by the Gibbs measure. In handling a nonlinearity of an arbitrary high degree, we make use of the Hermite polynomials and the white noise functional.

Keywords

generalized KdV equation, Gibbs measure, Hermite polynomial, white noise functional

2010 Mathematics Subject Classification

35Q53

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