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arXiv · 2412.08630

On growth of Sobolev norms for periodic nonlinear Schr\"{o}dinger and generalised Korteweg-de Vries equations under critical Gibbs dynamics

Fabian Höfer·Niko A. Nikov
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Abstract

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.

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Fabian Höfer, Niko A. Nikov. 2024-12-11. On growth of Sobolev norms for periodic nonlinear Schr\"{o}dinger and generalised Korteweg-de Vries equations under critical Gibbs dynamics. https://doi.org/10.1090/proc%2F17379

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