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 $T^{D}$ . The Hamiltonian $H=\int_{{\bf T}^D} \Vert\nabla u\Vert^2

\int_{{\bf T}^D} \vert u\vert^p $h a s c a n o n i c a l e q u a t i o n s w i t h s o l u t i o n s i n$ \Omega_N={ u\in L^2({\bf T}^D) :\int \vert u\vert^2\leq N} $. F o r$ D=1 $a n d$ 2\leq p<6 $,$ \Omega_N $s u p p o r t s t h e G i b b s m e a s u r e$ \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 $(Ω_{N}, ∥ \cdot ∥_{L^{2}}, ν)$ is a metric probability space of finite diameter that satisfies the logarithmic Sobolev inequalities for the periodic $K d V$ , the focussing cubic nonlinear Schr"odinger 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 $(Ω, ∥ \cdot ∥_{H^{- s}}, ν)$ which satisfies $L S I$ . In the above cases, $(Ω, d, ν)$ is the limit in $L^{2}$ transportation distance of finite-dimensional $(\Omega_n, \Vert \cdot \Vert,\nu_n)$ given by Fourier sums.

This paper concerns Gibbs measures $ν$ for some nonlinear PDE over the $D$ -torus $T^{D}$ . The Hamiltonian $H=\int_{{\bf T}^D} \Vert\nabla u\Vert^2

\int_{{\bf T}^D} \vert u\vert^p $h a s c a n o n i c a l e q u a t i o n s w i t h s o l u t i o n s i n$ \Omega_N={ u\in L^2({\bf T}^D) :\int \vert u\vert^2\leq N} $. F o r$ D=1 $a n d$ 2\leq p<6 $,$ \Omega_N $s u p p o r t s t h e G i b b s m e a s u r e$ \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 $(Ω_{N}, ∥ \cdot ∥_{L^{2}}, ν)$ is a metric probability space of finite diameter that satisfies the logarithmic Sobolev inequalities for the periodic $K d V$ , the focussing cubic nonlinear Schr"odinger 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 $(Ω, ∥ \cdot ∥_{H^{- s}}, ν)$ which satisfies $L S I$ . In the above cases, $(Ω, d, ν)$ is the limit in $L^{2}$ transportation distance of finite-dimensional $(\Omega_n, \Vert \cdot \Vert,\nu_n)$ given by Fourier sums.

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