The Equations of Navier-Stokes and Abstract Parabolic Equations

Wahl, Wolf von

doi:10.1007/978-3-663-13911-9

Cited by 217 publications

(103 citation statements)

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“…Two almost-up-to-date collections of results can be found in the review papers by Cannone [8] and von Wahl [32]. The solutions we find live in rather classical spaces, similar to those considered by Majda and Bertozzi [28] (mainly for the Euler equation).…”

Section: However a Virtual Lagrangian Dynamic Of The Particles Of Thsupporting

confidence: 66%

A Probabilistic Representation for the Vorticity of a Three-Dimensional Viscous Fluid and for General Systems of Parabolic Equations

Busnello

Flandoli

Romito

2005

Proceedings of the Edinburgh Mathematical Society

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A probabilistic representation formula for general systems of linear parabolic equations, coupled only through the zero-order term, is given. On this basis, an implicit probabilistic representation for the vorticity in a three-dimensional viscous fluid (described by the Navier-Stokes equations) is carefully analysed, and a theorem of local existence and uniqueness is proved. The aim of the probabilistic representation is to provide an extension of the Lagrangian formalism from the non-viscous (Euler equations) to the viscous case. As an application, a continuation principle, similar to the Beale-Kato-Majda blow-up criterion, is proved.

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Section: However a Virtual Lagrangian Dynamic Of The Particles Of Thsupporting

confidence: 66%

A Probabilistic Representation for the Vorticity of a Three-Dimensional Viscous Fluid and for General Systems of Parabolic Equations

Busnello

Flandoli

Romito

2005

Proceedings of the Edinburgh Mathematical Society

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“…Substituting the above estimates (13), (17), (18), (19), (20), (21) and (22) into (12), it follows that…”

Section: Lemmamentioning

confidence: 99%

A New Regularity Criterion for the Navier-Stokes Equations in Terms of the Gradient of One Velocity Component

Zhou¹

2002

Methods and Applications of Analysis

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“…We give a rather general sufficient condition for the existence of a small solution for all times, we do not require the exterior forces to be conservative, nor the boundary values to be constant, and we do not need to confine ourselves to three dimensions. (For the incompressible case see, e.g., [12]. )…”

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confidence: 99%