# Almost sure well-posedness of the cubic nonlinear Schr\

@inproceedings{Colliander2009AlmostSW, title={Almost sure well-posedness of the cubic nonlinear Schr\}, author={James E. Colliander and Tadahiro Oh}, year={2009} }

We consider the Cauchy problem for the one-dimensional periodic cubic non- linear Schrodinger 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 , and global well-posedness for each s > − 1 12 .

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#### References

SHOWING 1-10 OF 46 REFERENCES

Remarks on nonlinear smoothing under randomization for the periodic KdV and the cubic Szegö equation

- Mathematics
- 2010

We consider Cauchy problems of some dispersive PDEs with random initial data. In particular, we construct local-in-time solutions to the mean-zero periodic KdV almost surely for the initial data in… Expand

A Priori Bounds for the 1D Cubic NLS in Negative Sobolev Spaces

- Mathematics
- 2006

Author(s): Koch, Herbert; Tataru, Daniel | Abstract: We consider the cubic Nonlinear Schrodinger Equation in one space dimension, either focusing or defocusing. We prove that the solutions satisfy… Expand

Random data Cauchy theory for supercritical wave equations I: local theory

- Mathematics
- 2008

We study the local existence of strong solutions for the cubic nonlinear wave equation with data in Hs(M), s<1/2, where M is a three dimensional compact Riemannian manifold. This problem is… Expand

On ill-posedness for the one-dimensional periodic cubic Schrodinger equation

- Physics, Mathematics
- 2008

We prove the ill-posedness in $ H^s(\T) $, $s<0$, of the periodic cubic Schr\"odinger equation in the sense that the flow-map is not continuous from $H^s(\T) $ into itself for any fixed $ t\neq 0 $.… Expand

POWER SERIES SOLUTION OF A NONLINEAR SCHR ¨ ODINGER EQUATION

- Mathematics
- 2006

A slightly modified variant of the cubic periodic one-dimensional non- linear Schrodinger equation is shown to be well-posed, in a relatively weak sense, in certain function spaces wider than L 2 .… Expand

Invariant measures for the nonlinear Schrödinger equation on the disc

- Mathematics
- 2006

We study Gibbs measures invariant under the flow of the NLS on the unit disc of $\R^2$. For that purpose, we construct the dynamics on a phase space of limited Sobolev regularity and a wighted Wiener… Expand

Instability of the periodic nonlinear Schrodinger equation

- Mathematics
- 2003

We study the periodic non-linear Schrodinger equations with odd integer power nonlinearities, for initial data which are assumed to be small in some negative order Sobolev space, but which may have… Expand

An instability property of the nonlinear Schrödinger equation on $S^{d}$

- Mathematics
- 2002

We consider the NLS on spheres. We describe the nonlinear evolutions of the highest weight spherical harmonics. As a consequence, in contrast with the flat torus, we obtain that the flow map fails to… Expand

Invariant measure for a three dimensional nonlinear wave equation

- Mathematics
- 2007

We study the long time behavior of the subcritical (subcubic) defocussing nonlinear wave equation on the three dimensional ball, for random data of low regularity. We prove that for a large set of… Expand

Invariant Gibbs Measures and a.s. Global Well-Posedness for Coupled KdV Systems

- Mathematics
- 2009

We continue our study of the well-posedness theory of a one-parameter family of coupled KdV-type systems in the periodic setting. When the value of a coupling parameter \alpha \in (0, 4) \setminus 1,… Expand