Gibbs measure for the focusing fractional NLS on the torus

Rui Liang; Yuzhao Wang

doi:10.1137/21M1445946

Gibbs measure for the focusing fractional NLS on the torus

Rui Liang, Yuzhao Wang

Mathematics

Research output: Contribution to journal › Article › peer-review

1 Citation (Scopus)

Abstract

We study the construction of the Gibbs measures for the focusing mass-critical fractional nonlinear Schrödinger equation on the multidimensional torus. We identify the sharp mass threshold for normalizability and nonnormalizability of the focusing Gibbs measures, which generalizes the influential works of Lebowitz, Rose, and Speer [J. Statist. Phys., 50 (1988), pp. 657–687], Bourgain [Comm. Math. Phys., 166 (1994), pp. 1–26], and Oh, Sosoe, and Tolomeo [Invent. Math., 227 (2022), pp. 1323–1429] on the one-dimensional nonlinear Schrödinger equations. To this purpose, we establish an almost sharp fractional Gagliardo–Nirenberg–Sobolev inequality on the torus, which is of independent interest.

Original language	English
Pages (from-to)	6096-6118
Number of pages	23
Journal	SIAM Journal on Mathematical Analysis
Volume	54
Issue number	6
Early online date	17 Nov 2022
DOIs	https://doi.org/10.1137/21M1445946
Publication status	Published - Dec 2022

Bibliographical note

Funding Information:
Last Received by the editors September 13, 2021; accepted for publication (in revised form) July 5, 2022; published electronically November 17, 2022. https://doi.org/10.1137/21M1445946 Funding: The work of the second author was supported by the EPSRC New Investigator Award grant EP/V003178/1. \dagger School of Mathematics, Watson Building, University of Birmingham, Edgbaston, Birmingham, B15 2TT, United Kingdom ([email protected], [email protected]).

Publisher Copyright:
© 2022 Society for Industrial and Applied Mathematics.

Keywords

focusing Gibbs measure
fractional Gagliardo–Nirenberg–Sobolev inequality
fractional nonlinear Schrödinger equation
normalizability
variational approach

ASJC Scopus subject areas

Analysis
Computational Mathematics
Applied Mathematics

Access to Document

10.1137/21M1445946

Dynamics of singular stochastic nonlinear dispersive PDEs
Wang, Y. (Principal Investigator)
Engineering & Physical Science Research Council
1/04/21 → 31/03/24
Project: Research Councils

Cite this

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title = "Gibbs measure for the focusing fractional NLS on the torus",

abstract = "We study the construction of the Gibbs measures for the focusing mass-critical fractional nonlinear Schr{\"o}dinger equation on the multidimensional torus. We identify the sharp mass threshold for normalizability and nonnormalizability of the focusing Gibbs measures, which generalizes the influential works of Lebowitz, Rose, and Speer [J. Statist. Phys., 50 (1988), pp. 657–687], Bourgain [Comm. Math. Phys., 166 (1994), pp. 1–26], and Oh, Sosoe, and Tolomeo [Invent. Math., 227 (2022), pp. 1323–1429] on the one-dimensional nonlinear Schr{\"o}dinger equations. To this purpose, we establish an almost sharp fractional Gagliardo–Nirenberg–Sobolev inequality on the torus, which is of independent interest.",

keywords = "focusing Gibbs measure, fractional Gagliardo–Nirenberg–Sobolev inequality, fractional nonlinear Schr{\"o}dinger equation, normalizability, variational approach",

author = "Rui Liang and Yuzhao Wang",

note = "Funding Information: Last Received by the editors September 13, 2021; accepted for publication (in revised form) July 5, 2022; published electronically November 17, 2022. https://doi.org/10.1137/21M1445946 Funding: The work of the second author was supported by the EPSRC New Investigator Award grant EP/V003178/1. \dagger School of Mathematics, Watson Building, University of Birmingham, Edgbaston, Birmingham, B15 2TT, United Kingdom ([email protected], [email protected]). Publisher Copyright: {\textcopyright} 2022 Society for Industrial and Applied Mathematics.",

year = "2022",

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N1 - Funding Information: Last Received by the editors September 13, 2021; accepted for publication (in revised form) July 5, 2022; published electronically November 17, 2022. https://doi.org/10.1137/21M1445946 Funding: The work of the second author was supported by the EPSRC New Investigator Award grant EP/V003178/1. \dagger School of Mathematics, Watson Building, University of Birmingham, Edgbaston, Birmingham, B15 2TT, United Kingdom ([email protected], [email protected]). Publisher Copyright: © 2022 Society for Industrial and Applied Mathematics.

PY - 2022/12

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N2 - We study the construction of the Gibbs measures for the focusing mass-critical fractional nonlinear Schrödinger equation on the multidimensional torus. We identify the sharp mass threshold for normalizability and nonnormalizability of the focusing Gibbs measures, which generalizes the influential works of Lebowitz, Rose, and Speer [J. Statist. Phys., 50 (1988), pp. 657–687], Bourgain [Comm. Math. Phys., 166 (1994), pp. 1–26], and Oh, Sosoe, and Tolomeo [Invent. Math., 227 (2022), pp. 1323–1429] on the one-dimensional nonlinear Schrödinger equations. To this purpose, we establish an almost sharp fractional Gagliardo–Nirenberg–Sobolev inequality on the torus, which is of independent interest.

AB - We study the construction of the Gibbs measures for the focusing mass-critical fractional nonlinear Schrödinger equation on the multidimensional torus. We identify the sharp mass threshold for normalizability and nonnormalizability of the focusing Gibbs measures, which generalizes the influential works of Lebowitz, Rose, and Speer [J. Statist. Phys., 50 (1988), pp. 657–687], Bourgain [Comm. Math. Phys., 166 (1994), pp. 1–26], and Oh, Sosoe, and Tolomeo [Invent. Math., 227 (2022), pp. 1323–1429] on the one-dimensional nonlinear Schrödinger equations. To this purpose, we establish an almost sharp fractional Gagliardo–Nirenberg–Sobolev inequality on the torus, which is of independent interest.

KW - focusing Gibbs measure

KW - fractional Gagliardo–Nirenberg–Sobolev inequality

KW - fractional nonlinear Schrödinger equation

KW - normalizability

KW - variational approach

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VL - 54

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EP - 6118

JO - SIAM Journal on Mathematical Analysis

JF - SIAM Journal on Mathematical Analysis

IS - 6

ER -

Gibbs measure for the focusing fractional NLS on the torus

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

Access to Document

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Projects

Dynamics of singular stochastic nonlinear dispersive PDEs

Cite this