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

Abstract

The periodic KdV equation u_t=u_{xxx}+βuu_x arises from a Hamiltonian system with infinite-dimensional phase space L^2(T). Bourgain has shown that there exists a Gibbs measure νon balls \{ϕ:\VertΦ\Vert^2_{L^2}\leq N\} in the phase space such that the Cauchy problem for KdV is well posed on the support of ν, and νis invariant under the KdV flow. This paper shows that νsatisfies 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 periodic KdV equation and the cubic nonlinear Schrödinger equation, for small values of N.