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]).
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Communicated by J.L. Lebowitz
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Bourgain, J. Invariant measures for the2D-defocusing nonlinear Schrödinger equation. Commun.Math. Phys. 176, 421–445 (1996). https://doi.org/10.1007/BF02099556
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DOI: https://doi.org/10.1007/BF02099556