On the probabilistic well-posedness of the nonlinear Schrödinger equations with non-algebraic nonlinearities

Tadahiro Oh, Mamoru Okamoto, Oana Pocovnicu

Research output: Contribution to journalArticlepeer-review

Abstract

We consider the Cauchy problem for the nonlinear Schrödinger equations (NLS) with non-algebraic nonlinearities on the Euclidean space. In particular, we study the energy-critical NLS on ℝ^d, d=5,6, and energy-critical NLS without gauge invariance and prove that they are almost surely locally well-posed with respect to randomized initial data below the energy space. We also study the long time behavior of solutions to these equations: (i) we prove almost sure global well-posedness of the (standard) energy-critical NLS on ℝ^d, d=5,6, in the defocusing case, and (ii) we present a probabilistic construction of finite time blowup solutions to the energy-critical NLS without gauge invariance below the energy space.

Original languageEnglish
Pages (from-to)3479-3520
Number of pages44
JournalDiscrete and Continuous Dynamical Systems - Series A
Volume39
Issue number6
Early online dateFeb 2019
DOIs
Publication statusPublished - Jun 2019

Keywords

  • nonlinear Schrödinger equation
  • almost sure local well-posedness
  • almost sure global well-posedness
  • finite time blowup

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