Blower, Gordon
(2016)
*Concentration of the invariant measure for the periodic Zakharov, KdV and cubic NLS equations in 1D and 2D.*
Journal of Mathematical Analysis and Applications, 438 (1).
pp. 240-266.
ISSN 0022-247X

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## Abstract

This paper concerns Gibbs measures ν for some nonlinear PDE over the D -torus TD. The Hamiltonian H=∫TD‖∇u‖2−∫TD|u|p has canonical equations with solutions in ΩN={u∈L2(TD):∫|u|2≤N}; this N is a parameter in quantum field theory analogous to the number of particles in a classical system. For D=1 and 2≤p<6, ΩN supports the Gibbs measure ν(du)=Z−1e−H(u)∏x∈Tdu(x) which is normalized and formally invariant under the flow generated by the PDE. The paper proves that (ΩN,‖⋅‖L2,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 ΩN, a logarithmic Sobolev inequality also holds in the critical case p=6. For D=2, the Gross–Piatevskii equation has H=∫T2‖∇u‖2−∫T2(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 L2 transportation distance of finite-dimensional (Ωn,‖⋅‖,νn) given by Fourier sums.