Higher order expansions for the probabilistic local Cauchy theory of the cubic nonlinear Schrödinger equation on ℝ3

Árpád Bényi, Tadahiro Oh, Oana Pocovnicu

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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 languageEnglish
Pages (from-to)114-160
Number of pages47
JournalTransactions of the American Mathematical Society: Series B
Volume6
DOIs
Publication statusPublished - 4 Mar 2019

Keywords

  • Partial differential equations
  • Probability theory and stochastic processes

ASJC Scopus subject areas

  • Mathematics(all)

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