On the Navier–Stokes equation perturbed by rough transport noise

We consider the Navier–Stokes system in two and three space dimensions perturbed by transport noise and subject to periodic boundary conditions. The noise arises from perturbing the advecting velocity field by space–time-dependent noise that is smooth in space and rough in time. We study the system...

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Bibliographic Details
Published inJournal of evolution equations Vol. 19; no. 1; pp. 203 - 247
Main Authors Hofmanová, Martina, Leahy, James-Michael, Nilssen, Torstein
Format Journal Article
LanguageEnglish
Published Cham Springer International Publishing 08.03.2019
Springer Nature B.V
Subjects
Analysis
Boundary conditions
Fluid dynamics
Fluid flow
Mathematics
Mathematics and Statistics
Navier-Stokes equations
Noise
Time dependence
Transport
Velocity distribution
47J30
Rough paths
Navier–Stokes equation
35A15
Stochastic PDEs
Variational method
76D05
60H05
60H15
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Summary:We consider the Navier–Stokes system in two and three space dimensions perturbed by transport noise and subject to periodic boundary conditions. The noise arises from perturbing the advecting velocity field by space–time-dependent noise that is smooth in space and rough in time. We study the system within the framework of rough path theory and, in particular, the recently developed theory of unbounded rough drivers. We introduce an intrinsic notion of a weak solution of the Navier–Stokes system, establish suitable a priori estimates and prove existence. In two dimensions, we prove that the solution is unique and stable with respect to the driving noise.
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ISSN:1424-3199
1424-3202
DOI:10.1007/s00028-018-0473-z