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

Abstract

This paper concerns Gibbs measures $\hat{I}½$ for some nonlinear PDE over the $D$-torus ${\mathbf{T}}^D$. The Hamiltonian $H={\int }_{{\mathbf{T}}^D}\Vert \nabla u{\Vert }^2-{\int }_{{\mathbf{T}}^D}|u{|}^p$ has canonical equations with solutions in $\hat{I}{\text{c◯}}_N=\{u\in L^2({\mathbf{T}}^D):\int |u{|}^2\le N\}$. For $D=1$ and $2\le p<6$, $\hat{I}{\text{c◯}}_N$ supports the Gibbs measure $\hat{I}½(du)=Z^{-1}{e}^{-H(u)}{\prod }_{x\in \mathbf{T}}du(x)$ which is normalized and formally invariant under the flow generated by the PDE. The paper proves that $(\hat{I}{\text{c◯}}_N,\Vert \cdot {\Vert }_{L^2},\hat{I}½)$ 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 $\hat{I}{\text{c◯}}_N$, a logarithmic Sobolev inequality also holds in the critical case $p=6$. For $D=2$, the Gross--Piatevskii equation has $H={\int }_{{\mathbf{T}}^2}\Vert \nabla u{\Vert }^2-{\int }_{{\mathbf{T}}^2}(V\ast |u{|}^2)|u{|}^2$, for a suitable bounded interaction potential $V$ and the Gibbs measure $\hat{I}½$ lies on a metric probability space $(\hat{I}\text{c◯},\Vert \cdot {\Vert }_{H^{-s}},\hat{I}½)$ which satisfies $LSI$. In the above cases, $(\hat{I}\text{c◯},d,\hat{I}½)$ is the limit in $L^2$ transportation distance of finite-dimensional $(\hat{I}{\text{c◯}}_n,\Vert \cdot \Vert ,\hat{I}{½}_n)$ given by Fourier sums.

28 pages