### Abstract

We consider the cubic nonlinear Schrödinger equation (NLS) on ℝ3 with randomized initial data. In particular, we study an iterative approach based on a partial power series expansion in terms of the random initial data. By performing a fixed point argument around the second order expansion, we improve the regularity threshold for almost sure local well-posedness from our previous work. We further investigate a limitation of this iterative procedure. Finally, we introduce an alternative iterative approach, based on a modified expansion of arbitrary length, and prove almost sure local well-posedness of the cubic NLS in an almost optimal regularity range with respect to the original iterative approach based on a power series expansion.

Original language | English |
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Pages (from-to) | 114-160 |

Number of pages | 47 |

Journal | Transactions of the American Mathematical Society: Series B |

Volume | 6 |

DOIs | |

Publication status | Published - 4 Mar 2019 |

## Fingerprint Dive into the research topics of 'Higher order expansions for the probabilistic local Cauchy theory of the cubic nonlinear Schrödinger equation on ℝ<sup>3</sup>'. Together they form a unique fingerprint.

## Profiles

## Oana Pocovnicu

- School of Mathematical & Computer Sciences - Associate Professor
- School of Mathematical & Computer Sciences, Mathematics - Associate Professor

Person: Academic (Research & Teaching)