The periodic KdV equation arises from a Hamiltonian system with infinite-dimensional phase space L^2(T). Bourgain has shown that there exists a Gibbs measure \nu on balls in the phase space such that the Cauchy problem for KdV is well posed on the support of \nu, and \nu is invariant under the KdV flow. This paper shows that \nu satisfies a logarithmic Sobolev inequality. The stationary points of the Hamiltonian on spheres are found in terms of elliptic functions, and they are shown to be linearly stable. The paper also presents logarithmic Sobolev inequalities for the modified period KdV equation and the cubic nonlinear Schrodinger equation for small values of the number operator N.
The final, definitive version of this article has been published in the Journal, Stochastics: An International Journal of Probability and Stochastic Processes, 84 (4), 2012, Informa Plc