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

TL;DR

This paper establishes almost sure logarithmic growth bounds for Sobolev norms of solutions to focusing mass-critical NLS and gKdV equations on the torus, under the critical Gibbs measure, extending Bourgain's invariant measure approach.

Contribution

It provides the first almost sure growth bounds for Sobolev norms under the critical Gibbs measure for these equations, using a generalized invariant measure argument.

Findings

Sobolev norms grow at most logarithmically in time

Results hold for initial data in H^s with s<1/2

Applicable to focusing mass-critical NLS and gKdV equations

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.