On growth of Sobolev norms for periodic nonlinear Schrödinger and generalised Korteweg-de Vries equations under critical Gibbs dynamics
Fabian Höfer, Niko A. Nikov
TL;DR
The paper proves almost-sure logarithmic growth bounds for Sobolev norms of focusing mass-critical NLS and gKdV on the torus under the optimal Gibbs measure, addressing a critical barrier in Bourgain's invariant-measure approach where the density is not in L^p at the threshold. It replaces deterministic conservation-based control with an Orlicz-space-based integrability framework, deploying a soliton-manifold decomposition, GNS-optimiser stability, and a change-of-variables technique to bound a weighted Gibbs integral. The main results establish sup_t ||u(t)||_{H^s} ≲ log(2+T) for s<1/2, and extend the methodology to gKdV, achieving analogous log-growth in Fourier-Lebesgue spaces and subsequent H^s bounds via embeddings. Together, these contributions illuminate how invariant-measure dynamics and soliton geometry yield sharp probabilistic growth control at the critical threshold, with implications for long-time behavior of dispersive equations under random initial data.
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.