Global Solutions for the Defocusing Cubic Nonlinear Schrödinger Equations on 2-sphere

Abstract

Liouville’s theorem, a landmark in ergodic theory, states that Lebesgue measure is invariant under the flow of Hamiltonian system. In this paper, we obtain a global existence of solutions for the defocusing cubic nonlinear Schrödinger equations on 2-sphere with initial data distributed according to Gibbs measure, which is invariant in some mild sense under the flow of Wick renormalized Hamiltonian. After modulating the manifold, we also improve the regularity of support space of Gibbs measure and establish the almost sure global well-posedness.

Keywords

Gibbs measure / Wick ordered Hamiltonian / almost sure global well-posedness / 35Q55 / 37L40 / 58J70

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