GLOBAL WELL-POSEDNESS OF THE 3D NAVIER-STOKES EQUATIONS PERTURBED BY A DETERMINISTIC VECTOR FIELD

• 成果类型: Article
• 署名作者: Flandoli, Franco; Hofmanova, Martina; Luo, Dejun; Nilssen, Torstein
• 署名单位: Scuola Normale Superiore di Pisa; University of Bielefeld; Chinese Academy of Sciences; Academy of Mathematics & System Sciences, CAS; University of Agder
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/21-AAP1740
• 发表日期: 2022
• 页码: 2568-2586

摘要

We are concerned with the problem of global well-posedness of the 3D Navier-Stokes equations on the torus with unitary viscosity. While a full answer to this question seems to be out of reach of the current techniques, we establish a regularization by a deterministic vector field. More precisely, we consider the vorticity form of the system perturbed by an additional transport type term. Such a perturbation conserves the enstrophy and therefore a priori it does not imply any smoothing. Our main result is a construction of a deterministic vector field v = v(t, x) which provides the desired regularization of the system and yields global well-posedness for large initial data outside arbitrary small sets. The proof relies on probabilistic arguments developed by Flandoli and Luo, tools from rough path theory by Hofmanova, Leahy and Nilssen and a new Wong-Zakai approximation result, which itself combines probabilistic and rough path techniques.