The periodic KdV equation $u_t=u_{xxx}+\beta uu_x$ arises from a Hamiltonian system with infinite-dimensional phase space $L^2({\bf T})$. Bourgain has shown that there exists a Gibbs probability measure $\nu$ on balls $\{\phi :\Vert \phi\Vert^2_{L^2}\leq N\}$ 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 talk will show that $\nu$ satisfies a logarithmic Sobolev inequality. The seminar presents logarithmic Sobolev inequalities for the modified periodic KdV equation and the cubic nonlinear Schr\"odinger equation. There will also be recent results from Blower, Brett and Doust regarding spectral concentration phenomena for Hill's equation.
