A logarithmic Sobolev inequality for the invariant measure of the periodic Korteweg--de Vries equation

TL;DR

This paper proves a logarithmic Sobolev inequality for the invariant Gibbs measure of the periodic KdV equation, establishing stability and extending results to related equations.

Contribution

It demonstrates a logarithmic Sobolev inequality for the Gibbs measure of the periodic KdV, a novel result linking measure invariance and functional inequalities in infinite dimensions.

Findings

Gibbs measure satisfies a logarithmic Sobolev inequality

Stationary points are elliptic functions and linearly stable

Inequalities extended to modified KdV and nonlinear Schrödinger equations

Abstract

The periodic KdV equation u_t=u_{xxx}+\beta uu_x 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 \{\phi :\Vert\Phi\Vert^2_{L^2}\leq N\} 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 periodic KdV equation and the cubic nonlinear Schr\"odinger equation, for small values of N.