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

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:438

Article

Blower, Gordon¹ 

[1] Univ Lancaster, Dept Math & Stat, Lancaster LA1 4YF, England

关键词: Gibbs measure;    Logarithmic Sobolev inequality;    Transportation;   

DOI  :  10.1016/j.jmaa.2016.01.080

来源: Elsevier

【 摘 要 】

This paper concerns Gibbs measures v for some nonlinear PDE over the D-torus T-D. The Hamiltonian H = integral(TD) parallel to del u parallel to(2) - integral(TD) vertical bar u vertical bar(p) has canonical equations with solutions in Omega(N) = {u is an element of L-2(T-D) : integral vertical bar u vertical bar(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, Omega(N) supports the Gibbs measure v(du) = Z(-1)e(-H(u)) Pi(x is an element of T) du(x) which is normalized and formally invariant under the flow generated by the PDE. The paper proves that (Omega(N), parallel to . parallel to(L2), v) is a metric probability space of finite diameter that satisfies the logarithmic Sobolev inequalities for the periodic KdV, the focussing cubic nonlinear Schrodinger 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 = integral(T2) parallel to del u parallel to(2) - integral(T2) (V*vertical bar u vertical bar(2))vertical bar u vertical bar(2), for a suitable bounded interaction potential V and the Gibbs measure v lies on a metric probability space (Omega, parallel to . parallel to(H-s), v) which satisfies LSI. In the above cases, Omega, d, v) is the limit in L-2 transportation distance of finite-dimensional (Omega(N), parallel to . parallel to, v(n)) given by Fourier sums. (C) 2016 Elsevier Inc. All rights reserved.

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