JournalsjemsVol. 14, No. 4pp. 1275–1330

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

  • Andrea R. Nahmod

    University of Massachusetts, Amherst, USA
  • Tadahiro Oh

    Princeton University, USA
  • Luc Rey-Bellet

    University of Massachusetts, Amherst, USA
  • Gigliola Staffilani

    Massachusetts Institute of Technology, Cambridge, USA
Invariant weighted Wiener measures and  almost sure global well-posedness  for the periodic derivative NLS cover
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Abstract

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 FLs,r(\T){\mathcal F}L^{s,r}(\T) with s12s \ge \frac{1}{2}, 2<r<42 < r < 4, (s1)r<1(s-1)r <-1 and scaling like H12ϵ(T),H^{\frac{1}{2}-\epsilon}(\mathbb T), for small ϵ>0\epsilon >0. We also show the invariance of this measure.

Cite this article

Andrea R. Nahmod, Tadahiro Oh, Luc Rey-Bellet, Gigliola Staffilani, Invariant weighted Wiener measures and almost sure global well-posedness for the periodic derivative NLS. J. Eur. Math. Soc. 14 (2012), no. 4, pp. 1275–1330

DOI 10.4171/JEMS/333