On growth of Sobolev norms for periodic nonlinear Schrödinger and generalised Korteweg-de Vries equations under critical Gibbs dynamics
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866909972826161152 |
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| author | Höfer, Fabian Nikov, Niko A. |
| author_facet | Höfer, Fabian Nikov, Niko A. |
| contents | We prove logarithmic growth bounds on Sobolev norms of the focusing mass-critical NLS and gKdV equations on the torus, which hold almost surely under the focusing Gibbs measure with optimal mass threshold constructed by Oh, Sosoe, and Tolomeo [Invent. Math. 227 (2022), no. 3, 1323--1429]. More precisely, we will establish almost sure growth bounds for solutions $u(t)$ of the equations of the form \[ \sup_{t \in [-T,T]} \lVert u(t) \rVert_{H^s(\mathbb{T})} \lesssim_{s, u_0} \log(2+T)\] with initial data $u_0 \in H^s(\mathbb{T})$ for $s< \frac{1}{2}$. The proof uses a generalisation of Bourgain's invariant measure argument for measures in a suitable Orlicz space. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_08630 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On growth of Sobolev norms for periodic nonlinear Schrödinger and generalised Korteweg-de Vries equations under critical Gibbs dynamics Höfer, Fabian Nikov, Niko A. Analysis of PDEs Probability 35Q55 (Primary) 35Q53, 35R60 (Secondary) We prove logarithmic growth bounds on Sobolev norms of the focusing mass-critical NLS and gKdV equations on the torus, which hold almost surely under the focusing Gibbs measure with optimal mass threshold constructed by Oh, Sosoe, and Tolomeo [Invent. Math. 227 (2022), no. 3, 1323--1429]. More precisely, we will establish almost sure growth bounds for solutions $u(t)$ of the equations of the form \[ \sup_{t \in [-T,T]} \lVert u(t) \rVert_{H^s(\mathbb{T})} \lesssim_{s, u_0} \log(2+T)\] with initial data $u_0 \in H^s(\mathbb{T})$ for $s< \frac{1}{2}$. The proof uses a generalisation of Bourgain's invariant measure argument for measures in a suitable Orlicz space. |
| title | On growth of Sobolev norms for periodic nonlinear Schrödinger and generalised Korteweg-de Vries equations under critical Gibbs dynamics |
| topic | Analysis of PDEs Probability 35Q55 (Primary) 35Q53, 35R60 (Secondary) |
| url | https://arxiv.org/abs/2412.08630 |