Abstract
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 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 Hofmanová, Leahy and Nilssen and a new Wong–Zakai approximation result, which itself combines probabilistic and rough path techniques.
Funding Statement
M.H. gratefully acknowledges the financial support by the German Science Foundation DFG via the Collaborative Research Center SFB 1283 and the Research Unit FOR 2402. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant agreement No. 949981).
D.L. would like to thank the financial supports of the National Key R&D Program of China (No. 2020YFA0712700) and the National Natural Science Foundation of China (Nos. 11688101, 11931004, 12090014) and the Youth Innovation Promotion Association, CAS (No. 2017003).
Acknowledgements
We are very grateful to the referees for reading carefully the paper and for many valuable comments which helped us with improving the paper.
Citation
Franco Flandoli. Martina Hofmanová. Dejun Luo. Torstein Nilssen. "Global well-posedness of the 3D Navier–Stokes equations perturbed by a deterministic vector field." Ann. Appl. Probab. 32 (4) 2568 - 2586, August 2022. https://doi.org/10.1214/21-AAP1740
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