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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

Citations:

Zbl 0822.35126
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References:

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