Blower, Gordon
(2016)
*Concentration of the invariant measure for the periodic Zakharov, KdV and cubic NLS equations in 1D and 2D.*
Working Paper.
UNSPECIFIED.
(In Press)

## Abstract

This paper concerns metric probability spaces of random Fourier series which produce Gibbs measures 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 which give solutions of the PDE in $\Omega_N=\{ u\in L^2({\bf T}^D) :\int \vert u\vert^2\leq N\}$. Also $\Omega_N$ supports the Gibbs measure $\nu_N(du)=Z^{-1}e^{-H(u)}\prod_{x\in {\bf T}^D} du(x)$ which is normalized and formally invariant under the flow generated by the PDE. For $D=1$, the paper proves that $(\Omega_N, \Vert\cdot\Vert_{L^2}, \nu_N)$ 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 Hamiltonian with $U(u)=\int_{{\bf T}^2} (V\ast \vert u\vert^2 )\vert u\vert^2$, for a suitable bounded interaction potential $V$ and the Gibbs measure $\nu$ lies on a metric probability space $(\Omega , \Vert\cdot\Vert_{H^{-s}}, \nu )$ of finite diameter which satisfies $LSI$.