## Invariant measures for the 2D-defocusing nonlinear Schrödinger equation.(English)Zbl 0852.35131

Summary: Consider the 2D defocusing cubic NLS $$iu_t+ \Delta u- u|u|^2= 0$$ with Hamiltonian $$\int (|\nabla \phi|^2+ {1\over 2} |\phi|^4)$$. It is shown that the Gibbs measure constructed from the Wick ordered Hamiltonian, i.e. replacing $$|\phi|^4$$ by $$:|\phi|^4:$$, is an invariant measure for the appropriately modified equation $iu_t+ \Delta u- [u|u|^2- 2(\int |u|^2 dx)u]= 0.$ There is a well defined flow on the support of the measure. In fact, it is shown that for almost all data $$\phi$$ the solution $$u$$, $$u(0)= \phi$$, satisfies $$u(t)- e^{it \Delta} \phi\in C_{H^s} (\mathbb{R})$$, for some $$s> 0$$. 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. ibid. 166, No. 1, 1-26 (1994; Zbl 0822.35126)].

### MSC:

 35Q55 NLS equations (nonlinear Schrödinger equations) 37C10 Dynamics induced by flows and semiflows

Zbl 0822.35126
Full Text:

### References:

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