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 υ = υ(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 Hofmanová, Leahy and Nilssen and a new Wong-Zakai approximation result, which itself combines probabilistic and rough path techniques.

Global well-posedness of the 3D Navier–Stokes equations perturbed by a deterministic vector field

Flandoli, Franco
;
2022

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 υ = υ(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 Hofmanová, Leahy and Nilssen and a new Wong-Zakai approximation result, which itself combines probabilistic and rough path techniques.
2022
Settore MAT/06 - Probabilita' e Statistica Matematica
3D Navier-Stokes equations; regularization by noise; vorticity form; well-posedness; Wong-Zakai principle
   Mathematical analysis of fluid flows: the challenge of randomness
   FluFloRan
   European Commission
   Horizon 2020 Framework Programme
   949981
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11384/140270
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