Statistical mechanics of the periodic Benjamin–Ono equation

Abstract

The periodic Benjamin-Ono equation is an autonomous Hamiltonian system with a Gibbs measure on L 2 (T). The paper shows that the Gibbs measures on bounded balls of L 2 satisfy some logarithmic Sobolev inequalities. The space of n-soliton solutions of the periodic Benjamin-Ono equation, as discovered by Case, is a Hamiltonian system with an invariant Gibbs measure. As n → ∞, these Gibbs measures exhibit a concentration of measure phenomenon. Case introduced soliton solutions that are parameterised by atomic measures in the complex plane. The limiting distributions of these measures gives the density of a compressible gas that satisfies the isentropic Euler equations.Date: 1st February 2019.