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Invariant weighted Wiener measures and almost sure global well-posedness for the periodic derivative NLS

Andrea R. NahmodTadahiro OhLuc Rey-BelletGigliola Staffilani — 2012

Journal of the European Mathematical Society

We construct an invariant weighted Wiener measure associated to the periodic derivative nonlinear Schrödinger equation in one dimension and establish global well-posedness for data living in its support. In particular almost surely for data in a Fourier–Lebesgue space L s , r ( T ) with s 1 2 , 2 < r < 4 , ( s - 1 ) r < - 1 and scaling like H 1 2 - ϵ ( 𝕋 ) , for small ϵ > 0 . We also show the invariance of this measure.

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