Invariance of the Gibbs measures for periodic generalized Korteweg-de Vries equations

TL;DR

This paper proves the invariance of Gibbs measures for periodic generalized Korteweg-de Vries equations with high nonlinearities by establishing well-posedness in Fourier-Lebesgue spaces and applying Bourgain's invariant measure method.

Contribution

It extends Bourgain's invariance results to gKdV equations with quartic or higher nonlinearities using Fourier-Lebesgue space techniques.

Findings

Constructed almost sure global solutions for gauged gKdV.

Proved invariance of Gibbs measures under the flow.

Established well-posedness in Fourier-Lebesgue spaces.

Abstract

In this paper, we study the Gibbs measures for periodic generalized Korteweg-de Vries equations (gKdV) with quartic or higher nonlinearities. In order to bypass the analytical ill-posedness of the equation in the Sobolev support of the Gibbs measures, we establish deterministic well-posedness of the gauged gKdV equations within the framework of the Fourier-Lebesgue spaces. Our argument relies on bilinear and trilinear Strichartz estimates adapted to the Fourier-Lebesgue setting. Then, following Bourgain's invariant measure argument, we construct almost sure global-in-time dynamics and show invariance of the Gibbs measures for the gauged equations. These results can be brought back to the ungauged side by inverting the gauge transformation and exploiting the invariance of the Gibbs measures under spatial translations. We thus complete the program initiated by Bourgain (1994) on the invariance of the Gibbs measures for periodic gKdV equations.