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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 |
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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 | |
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
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Dive into the research topics of 'Gibbs measure for the focusing fractional NLS on the torus'. Together they form a unique fingerprint.Projects
- 1 Finished
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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