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arXiv · 0904.2820

Almost sure well-posedness of the cubic nonlinear Schrödinger equation below L^2(T)

James Colliander·Tadahiro Oh
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Abstract

We consider the Cauchy problem for the one-dimensional periodic cubic nonlinear Schrödinger equation (NLS) with initial data below L^2. In particular, we exhibit nonlinear smoothing when the initial data are randomized. Then, we prove local well-posedness of NLS almost surely for the initial data in the support of the canonical Gaussian measures on H^s(T) for each s > -1/3, and global well-posedness for each s > -1/12.

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James Colliander, Tadahiro Oh. 2011-08-18. Almost sure well-posedness of the cubic nonlinear Schrödinger equation below L^2(T). https://doi.org/10.1215/00127094-1507400

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