Concentration of the invariant measure for the periodic Zakharov, KdV and cubic NLS equations in 1D and 2D

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 $LSI$.