A logarithmic Sobolev inequality for the invariant measure of the periodic Korteweg–de Vries equation

Abstract

The periodic Korteweg–de Vries (KdV) equation arises from a Hamiltonian system with infinite-dimensional phase space . Bourgain has shown that there exists a Gibbs probability measure ν on balls in the phase space such that the Cauchy problem for KdV is well posed on the support of ν, and ν is invariant under the KdV flow. This paper shows that ν satisfies a logarithmic Sobolev inequality. This paper also presents logarithmic Sobolev inequalities for the modified periodic KdV equation and the cubic nonlinear Schrödinger equation, for small values of N.