Concentration of the invariant measures for the periodic Zakharov, KdV, NLS and Gross--Piatevskii equations in 1D and 2D
This paper concerns Gibbs measures $ν$ 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 $Ω_N=\{ u\in L^2({\bf T}^D) :\int \vert u\vert^2\leq N\}$. For $D=1$ and $2\leq p<6$, $Ω_N$ supports the Gibbs measure $ν(du)=Z^{-1}e^{-H(u)}\prod_{x\in {\bf T}} du(x)$ which is normalized and formally invariant under the flow generated by the PDE. The paper proves that $(Ω_N, \Vert\cdot\Vert_{L^2}, ν)$ is a metric probability space of finite diameter that satisfies the logarithmic Sobolev inequalities for the periodic $KdV$, the focussing cubic nonlinear Schrödinger equation and the periodic Zakharov system. For suitable subset of $Ω_N$, 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 $ν$ lies on a metric probability space $(Ω, \Vert\cdot\Vert_{H^{-s}}, ν)$ which satisfies $LSI$. In the above cases, $(Ω, d, ν)$ is the limit in $L^2$ transportation distance of finite-dimensional $(Ω_n, \Vert \cdot \Vert,ν_n)$ given by Fourier sums.