  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.
