Concentration of the invariant measures for the periodic Zakharov, KdV, NLS and Gross--Piatevskii equations in 1D and 2D

TL;DR

This paper studies the concentration properties of invariant Gibbs measures for various nonlinear PDEs on tori in 1D and 2D, establishing logarithmic Sobolev inequalities and measure invariance in infinite-dimensional settings.

Contribution

It proves that Gibbs measures for several key PDEs support logarithmic Sobolev inequalities and are limits of finite-dimensional measures, advancing understanding of measure concentration in infinite dimensions.

Findings

Gibbs measures are supported on finite-diameter metric spaces.

Logarithmic Sobolev inequalities hold for these measures.

Measures are limits of finite-dimensional Fourier sum measures.

Abstract

This paper concerns Gibbs measures $\nu$ for some nonlinear PDE over the $D$-torus ${\bf T}^D$. The Hamiltonian $H=\int_{{\bf T}^D} \Vert\nabla u\Vert^2 - \int_{{\bf T}^D} \vert u\vert^p$ has canonical equations with solutions in $\Omega_N=\{ u\in L^2({\bf T}^D) :\int \vert u\vert^2\leq N\}$. For $D=1$ and $2\leq p<6$, $\Omega_N$ supports the Gibbs measure $\nu (du)=Z^{-1}e^{-H(u)}\prod_{x\in {\bf T}} du(x)$ which is normalized and formally invariant under the flow generated by the PDE. The paper proves that $(\Omega_N, \Vert\cdot\Vert_{L^2}, \nu )$ is a metric probability space of finite diameter that satisfies the logarithmic Sobolev inequalities for the periodic $KdV$, the focussing cubic nonlinear Schr\"odinger equation and the periodic Zakharov system. For suitable subset of $\Omega_N$, a logarithmic Sobolev inequality also holds in the critical case $p=6$. For $D=2$, the Gross--Piatevskii equation has $H=\int_{{\bf T}^2} \Vert\nabla u\Vert^2-\int_{{\bf T}^2} (V\ast \vert u\vert^2 ) \vert u\vert^2$, for a suitable bounded interaction potential $V$ and the Gibbs measure $\nu$ lies on a metric probability space $(\Omega , \Vert\cdot\Vert_{H^{-s}}, \nu )$ which satisfies $LSI$. In the above cases, $(\Omega, d, \nu )$ is the limit in $L^2$ transportation distance of finite-dimensional $(\Omega_n, \Vert \cdot \Vert,\nu_n)$ given by Fourier sums.