Duke Mathematical Journal

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

James Colliander and Tadahiro Oh

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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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Duke Math. J., Volume 161, Number 3 (2012), 367-414.

First available in Project Euclid: 1 February 2012

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Zentralblatt MATH identifier

Primary: 35Q55: NLS-like equations (nonlinear Schrödinger) [See also 37K10]
Secondary: 37K05: Hamiltonian structures, symmetries, variational principles, conservation laws 37L50: Noncompact semigroups; dispersive equations; perturbations of Hamiltonian systems 37L40: Invariant measures


Colliander, James; Oh, Tadahiro. Almost sure well-posedness of the cubic nonlinear Schrödinger equation below $L^{2}(\mathbb{T})$. Duke Math. J. 161 (2012), no. 3, 367--414. doi:10.1215/00127094-1507400. https://projecteuclid.org/euclid.dmj/1328105284

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