April 2021 On a rough perturbation of the Navier–Stokes system and its vorticity formulation
Martina Hofmanová, James-Michael Leahy, Torstein Nilssen
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Ann. Appl. Probab. 31(2): 736-777 (April 2021). DOI: 10.1214/20-AAP1603


We introduce a rough perturbation of the Navier–Stokes system and justify its physical relevance from balance of momentum and conservation of circulation in the inviscid limit. We present a framework for a well-posedness analysis of the system. In particular, we define an intrinsic notion of strong solution based on ideas from the rough path theory and study the system in an equivalent vorticity formulation. In two space dimensions, we prove that well-posedness and enstrophy balance holds. Moreover, we derive rough path continuity of the equation, which yields a Wong–Zakai result for Brownian driving paths, and show that for a large class of driving signals, the system generates a continuous random dynamical system. In dimension three, the noise is not enstrophy balanced, and we establish the existence of local in time solutions.


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Martina Hofmanová. James-Michael Leahy. Torstein Nilssen. "On a rough perturbation of the Navier–Stokes system and its vorticity formulation." Ann. Appl. Probab. 31 (2) 736 - 777, April 2021. https://doi.org/10.1214/20-AAP1603


Received: 1 April 2019; Revised: 1 January 2020; Published: April 2021
First available in Project Euclid: 1 April 2021

Digital Object Identifier: 10.1214/20-AAP1603

Primary: 35A15 , 47J30 , 60H05 , 60H15 , 76D05

Keywords: Navier–Stokes system , Rough paths , stochastic PDEs , variational method

Rights: Copyright © 2021 Institute of Mathematical Statistics


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Vol.31 • No. 2 • April 2021
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