A Remark on the Log-Sobolev Inequality for the Gibbs Measure of the Focusing Schrödinger Equation

Guopeng Li; Jiawei Li; Leonardo Tolomeo

doi:10.1007/s10884-026-10509-y

A Remark on the Log-Sobolev Inequality for the Gibbs Measure of the Focusing Schrödinger Equation

Guopeng Li
, Jiawei Li
, Leonardo Tolomeo^*

^*Corresponding author for this work

Beijing Institute of Technology
The University of Edinburgh and The Maxwell Institute for the Mathematical Sciences

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

Abstract

We consider the question of showing a log-Sobolev inequality for the Gibbs measure of the focusing Schrödinger equation built by Lebowitz-Rose-Speer (1988), formally given by (Formula presented.) When 2≤p≤4, we show that these measures indeed satisfy a log-Sobolev inequality. When p>4, we establish a lower bound for the Hessian of the effective potential. This implies that the known convexity-based multiscale techniques for the log-Sobolev inequalities cannot be applied to the measure ρ.

Original language	English
Journal	Journal of Dynamics and Differential Equations
DOIs	https://doi.org/10.1007/s10884-026-10509-y
Publication status	Accepted/In press - 2026
Externally published	Yes

Keywords

Gibbs measure
Logarithmic Sobolev inequalities
Nonlinear Schrödinger equation

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10.1007/s10884-026-10509-y

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title = "A Remark on the Log-Sobolev Inequality for the Gibbs Measure of the Focusing Schr{\"o}dinger Equation",

abstract = "We consider the question of showing a log-Sobolev inequality for the Gibbs measure of the focusing Schr{\"o}dinger equation built by Lebowitz-Rose-Speer (1988), formally given by (Formula presented.) When 2≤p≤4, we show that these measures indeed satisfy a log-Sobolev inequality. When p>4, we establish a lower bound for the Hessian of the effective potential. This implies that the known convexity-based multiscale techniques for the log-Sobolev inequalities cannot be applied to the measure ρ.",

keywords = "Gibbs measure, Logarithmic Sobolev inequalities, Nonlinear Schr{\"o}dinger equation",

author = "Guopeng Li and Jiawei Li and Leonardo Tolomeo",

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year = "2026",

doi = "10.1007/s10884-026-10509-y",

language = "English",

journal = "Journal of Dynamics and Differential Equations",

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T1 - A Remark on the Log-Sobolev Inequality for the Gibbs Measure of the Focusing Schrödinger Equation

AU - Li, Guopeng

AU - Li, Jiawei

AU - Tolomeo, Leonardo

N1 - Publisher Copyright: © The Author(s) 2026.

PY - 2026

Y1 - 2026

N2 - We consider the question of showing a log-Sobolev inequality for the Gibbs measure of the focusing Schrödinger equation built by Lebowitz-Rose-Speer (1988), formally given by (Formula presented.) When 2≤p≤4, we show that these measures indeed satisfy a log-Sobolev inequality. When p>4, we establish a lower bound for the Hessian of the effective potential. This implies that the known convexity-based multiscale techniques for the log-Sobolev inequalities cannot be applied to the measure ρ.

AB - We consider the question of showing a log-Sobolev inequality for the Gibbs measure of the focusing Schrödinger equation built by Lebowitz-Rose-Speer (1988), formally given by (Formula presented.) When 2≤p≤4, we show that these measures indeed satisfy a log-Sobolev inequality. When p>4, we establish a lower bound for the Hessian of the effective potential. This implies that the known convexity-based multiscale techniques for the log-Sobolev inequalities cannot be applied to the measure ρ.

KW - Gibbs measure

KW - Logarithmic Sobolev inequalities

KW - Nonlinear Schrödinger equation

UR - https://www.scopus.com/pages/publications/105037472085

U2 - 10.1007/s10884-026-10509-y

DO - 10.1007/s10884-026-10509-y

M3 - Article

AN - SCOPUS:105037472085

SN - 1040-7294

JO - Journal of Dynamics and Differential Equations

JF - Journal of Dynamics and Differential Equations

ER -

A Remark on the Log-Sobolev Inequality for the Gibbs Measure of the Focusing Schrödinger Equation

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Keywords

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