A remark on the log-Sobolev inequality for the Gibbs measure of the focusing Schr\"odinger equation

TL;DR

This paper investigates the log-Sobolev inequality for the Gibbs measure associated with the focusing Schrödinger equation, establishing its validity for certain nonlinearities and identifying limitations for others.

Contribution

It proves the log-Sobolev inequality for the Gibbs measure when 2 ≤ p ≤ 4 and shows that existing techniques fail for p > 4 due to the Hessian's properties.

Findings

Log-Sobolev inequality holds for 2 ≤ p ≤ 4.

Techniques cannot establish the inequality for p > 4 due to Hessian bounds.

Provides insights into the measure's geometric properties.

Abstract

We consider the question of showing a log-Sobolev inequality for the Gibbs measure of the focusing Schr\"odinger equation built by Lebowitz-Rose-Speer (1988), formally given by $$ d\rho \propto \exp\big(\frac 1 p\int_{\mathbb T} |u|^p d x - \frac 12\int_{\mathbb T} |\nabla u|^2 d x - \frac 12\int_{\mathbb T} |u|^2 d x\big) \mathbf 1_{\| u \|_{L^2(\mathbb T)}^2 \le K}dud\overline{u}. $$ When $2 \le p \le 4$, we show that these measures indeed satisfy a log-Sobolev inequality. When $p> 4$, we show a lower bound for the Hessian of the potential, which implies that the known techniques to show these inequalities cannot apply to the measure $\rho$.