Three-dimensional Navier-Stokes equations for singular filament initial data
type: Item
instance of: scholarly article
title: Three-dimensional Navier-Stokes equations for singular filament initial data
zbMATH Open document ID: 0675.76072
DOI: 10.1016/0022-0396(88)90004-6
author: S. H. Smith
published in: Journal of Differential Equations
publication date: 1988-01-01T00:00:00Z
review text: A slightly altered form of the traditional Navier-Stokes equations can be written in which, instead of the velocity vector \(\vec u\), the vorticity \({\vec \omega}\) (\({\vec \omega}=curl \vec u)\) is treated as unknown. This transformation is particularly useful when the vortical motions in a viscous fluid must be studied. In particular, it may happen that vortices are initially active on an element of a curve, and consequently, the system of equations of Navier-Stokes type describing this motion possesses a singular initial datum, analytically characterized by a measure located on the initial filament. The central assumptions to treating the problem is that, at any subsequent time, the instantaneous filament is diffeomorphic to the initial filament. This assumption being made, the so called mollified problem is considered, in which, instead of the originary quantities \(\vec u\), \({\vec \omega}\), their time convolutions, denoted by \(\vec u^{\epsilon}\), \({\vec \omega}^{\epsilon}\), are introduced. The system in terms of the new unknowns admits a unique classical solution, for which an a priori bound, independent of \(\epsilon\), can be given. It follows that, by applying compactness arguments, each \(\vec u^{\epsilon}\) weakly converges to \(\vec u\) in \(L^{\infty}\). Moreover, the further assumption that the \(\vec u\) norm decreases allows the uniqueness of the solution to be proven.
reviewed by: Piero Villaggio
Mathematics Subject Classification ID: 76N10
Mathematics Subject Classification ID: 35Q30
zbMATH DE Number: 4106495
zbMATH Keywords: Navier-Stokes equations
zbMATH Keywords: vortices
zbMATH Keywords: instantaneous filament
zbMATH Keywords: classical solution
zbMATH Keywords: compactness arguments
MaRDI profile type: Publication
full work available at URL: 0022-0396(88)90004-6
OpenAlex ID: W1987676723
cites work: Planar Navier-Stokes flow for singular initial data
cites work: Partial regularity of suitable weak solutions of the navier-stokes equations
cites work: Q3744590
cites work: Vortex Filament Method
cites work: Q4841248
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description: scientific article; zbMATH DE number 4106495