Almost sure well-posedness of the cubic nonlinear Schrödinger equation below L2(T)

J. Colliander, Tadahiro Oh

Type: Article

Publication Date: 2012-02-01

Citations: 153

DOI: https://doi.org/10.1215/00127094-1507400

Abstract

We consider the Cauchy problem for the 1-dimensional periodic cubic nonlinear Schrödinger (NLS) equation with initial data below L2. In particular, we exhibit nonlinear smoothing when the initial data are randomized. Then, we prove local well-posedness of the NLS equation almost surely for the initial data in the support of the canonical Gaussian measures on Hs(T) for each s>−13, and global well-posedness for each s>−112.

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