Invariant measures for the2D-defocusing nonlinear Schrödinger equation

Abstract

Consider the2D defocusing cubic NLSiu _t+Δu−u|u|²=0 with Hamiltonian$\smallint \left( {\left| {\nabla \phi } \right|^2 + \tfrac{1}{2}\left| \phi \right|^4 } \right)$. It is shown that the Gibbs measure constructed from the Wick ordered Hamiltonian, i.e. replacing |φ|⁴ by |φ|⁴ :, is an invariant measure for the appropriately modified equationiu _t + Δu‒ [u|u ²−2(∫|u|² dx)u]=0. There is a well defined flow on thesupport of the measure. In fact, it is shown that for almost all data ϕ the solutionu, u(0)=ϕ, satisfiesu(t)−e ^itΔφ ∈C _Hs (ℝ), for somes>0. First a result local in time is established and next measure invariance considerations are used to extend the local result to a global one (cf. [B2]).