A Remark on the Log-Sobolev Inequality for the Gibbs Measure of the Focusing Schrödinger Equation
Abstract
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 $$\begin{aligned} d\rho \propto \exp \big (\frac{1}{p}\int _{\mathbb T} |u|^p d x - \frac{1}{2}\int _{\mathbb T} |\nabla u|^2 d x - \frac{1}{2}\int _{\mathbb T} |u|^2 d x\big ) \mathbbm {1}_{\Vert u \Vert _{L^2(\mathbb T)}^2 \le K}dud\overline{u}. \end{aligned}$$ d ρ ∝ exp ( 1 p ∫ T | u | p d x - 1 2 ∫ T | ∇ u | 2 d x - 1 2 ∫ T | u | 2 d x ) 1 ‖ u ‖ L 2 ( T ) 2 ≤ K d u d u ¯ . When $$2 \le p \le 4$$ 2 ≤ p ≤ 4 , we show that these measures indeed satisfy a log-Sobolev inequality. When $$p> 4$$ 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 $$\rho $$ ρ .
Fields of Study & Subjects
Nonlinear Partial Differential Equations
Spectral Theory in Mathematical Physics
Advanced Mathematical Physics Problems
Publication details
- Journal / source
- Journal of Dynamics and Differential Equations
- Publisher
- Springer Science and Business Media LLC
- Volume
- 38
- Issue
- S1
- Pages
- 455-471
- Publication date
- 25 April 2026
- Language
- EN