Work

On growth of Sobolev norms for periodic nonlinear Schrödinger and generalised Korteweg-de Vries equations under critical Gibbs dynamics

Year: 2025

Type: article

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 ) u(t) of the equations of the form \[ sup t ∈ [ − T , T ] ‖ u ( t ) ‖ H s ( T ) ≲ s , u 0 log ⁡ ( 2 + T ) \sup _{t \in [-T,T]} \|u(t)\|_{H^s(\mathbb {T})} \lesssim _{s, u_0} \log (2+T) \] with initial data u 0 ∈ H s ( T ) u_0 \in H^s(\mathbb {T}) for s > 1 2 s> \frac {1}{2} . The proof uses a generalisation of Bourgain’s invariant measure argument for measures in a suitable Orlicz space.

Source: Proceedings of the American Mathematical Society

DOI: 10.1090/proc/17379

Authors: Fabian Höfer, Niko Nikov

Institutions: University of Münster, University of Edinburgh

Language: English

Topic: Advanced Mathematical Physics Problems

Subfield: Mathematical Physics

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

Funder: Deutsche Forschungsgemeinschaft

Awards: G1652504919, EXC 2044: Mathematik Münster: Dynamik – Geometrie – Struktur, G6938060324