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 ν for some nonlinear PDE over the D-torus u) x∈T du(x) which is normalized and formally invariant under the flow generated by the PDE. The paper proves that (Ω N , • 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 = T 2 ∇u 2 − T 2 (V * |u| 2 )|u| 2 , for a suitable bounded interaction potential V and the Gibbs measure ν lies on a metric probability space (Ω, • 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 , • , ν n ) given by Fourier sums.