Gibbs measure for the focusing fractional NLS on the torus

Rui Liang, Yuzhao Wang

Research output: Contribution to journalArticlepeer-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 languageEnglish
Pages (from-to)6096-6118
Number of pages23
JournalSIAM Journal on Mathematical Analysis
Volume54
Issue number6
Early online date17 Nov 2022
DOIs
Publication statusPublished - 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

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