Concentration of the invariant measure for the periodic Zakharov, KdV and cubic NLS equations in 1D and 2D
- Blower, Gordon
DOIAbstract
This paper concerns Gibbs measures ν for some nonlinear PDE over the D -torus TD. The Hamiltonian H=∫TD‖∇u‖2−∫TD|u|p has canonical equations with solutions in ΩN={u∈L2(TD):∫|u|2≤N}; this N is a parameter in quantum field theory analogous to the number of particles in a classical system. For D=1 and 2≤p<6, ΩN supports the Gibbs measure ν(du)=Z−1e−H(u)∏x∈Tdu(x) which is normalized and formally invariant under the flow generated by the PDE. The paper proves that (ΩN,‖⋅‖L2,ν) is a metric probability space of finite diameter that satisfies the logarithmic Sobolev inequalities for the periodic KdV , the focussing cubic nonlinear Schrödinger equation and the periodic Zakharov system. For suitable subset of ΩN, a logarithmic Sobolev inequality also...
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