A remark on the log-Sobolev inequality for the Gibbs measure of the focusing Schrödinger equation
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911300496392192 |
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| author | Li, Guopeng Li, Jiawei Tolomeo, Leonardo |
| author_facet | Li, Guopeng Li, Jiawei Tolomeo, Leonardo |
| contents | 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
$$ dρ\propto \exp\big(\frac 1 p\int_{\mathbb T} |u|^p d x - \frac 12\int_{\mathbb T} |\nabla u|^2 d x - \frac 12\int_{\mathbb T} |u|^2 d x\big) \mathbf 1_{\| u \|_{L^2(\mathbb T)}^2 \le K}dud\overline{u}. $$
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 $ρ$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_03897 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A remark on the log-Sobolev inequality for the Gibbs measure of the focusing Schrödinger equation Li, Guopeng Li, Jiawei Tolomeo, Leonardo Analysis of PDEs Probability 28C20, 35Q55, 60H30 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 $$ dρ\propto \exp\big(\frac 1 p\int_{\mathbb T} |u|^p d x - \frac 12\int_{\mathbb T} |\nabla u|^2 d x - \frac 12\int_{\mathbb T} |u|^2 d x\big) \mathbf 1_{\| u \|_{L^2(\mathbb T)}^2 \le K}dud\overline{u}. $$ 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 $ρ$. |
| title | A remark on the log-Sobolev inequality for the Gibbs measure of the focusing Schrödinger equation |
| topic | Analysis of PDEs Probability 28C20, 35Q55, 60H30 |
| url | https://arxiv.org/abs/2512.03897 |