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2022•SIAM Journal on Mathematical AnalysisRequires access

Gibbs Measure for the Focusing Fractional NLS on the Torus

Rui Liang, Yuzhao Wang

Open publisher page 7 citations

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.

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What this paper is about

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.

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Available 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.

Key concepts: Torus, Mathematics, Measure (data warehouse), Sobolev space, Gibbs measure, Nonlinear system, Mathematical analysis, Sobolev inequality

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