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

Tadahiro Oh; Mamoru Okamoto; Oana Pocovnicu

doi:10.3934/dcds.2019144

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

Tadahiro Oh, Mamoru Okamoto, Oana Pocovnicu

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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 language	English
Pages (from-to)	3479-3520
Number of pages	42
Journal	Discrete and Continuous Dynamical Systems-Series A
Volume	39
Issue number	6
Early online date	17 Mar 2019
DOIs	https://doi.org/10.3934/dcds.2019144
Publication status	Published - Jun 2019

Keywords

Almost sure global well-posedness
Almost sure local well-posedness
Finite time blowup
Non-algebraic nonlinearity
Nonlinear Schrodinger equation

ASJC Scopus subject areas

Analysis
Discrete Mathematics and Combinatorics
Applied Mathematics

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title = "On the probabilistic well-posedness of the nonlinear Schr{\"o}dinger equations with non-algebraic nonlinearities",

abstract = "We consider the Cauchy problem for the nonlinear Schr{\"o}dinger equations (NLS) with non-algebraic nonlinearities on the Euclidean space. In particular, we study the energy-critical NLS on Rd, 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 Rd, 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.",

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author = "Tadahiro Oh and Mamoru Okamoto and Oana Pocovnicu",

year = "2019",

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language = "English",

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publisher = "Southwest Missouri State University",

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AU - Pocovnicu, Oana

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N2 - 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 Rd, 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 Rd, 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.

AB - 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 Rd, 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 Rd, 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.

KW - Almost sure global well-posedness

KW - Almost sure local well-posedness

KW - Finite time blowup

KW - Non-algebraic nonlinearity

KW - Nonlinear Schrodinger equation

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JF - Discrete and Continuous Dynamical Systems-Series A

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

Abstract

Keywords

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