Lagrangian representation of a flow
type: Item
instance of: scholarly article
title: Lagrangian representation of a flow
zbMATH Open document ID: 0546.76039
DOI: 10.1016/0022-0396(85)90065-8
published in: Journal of Differential Equations
publication date: 1985-01-01T00:00:00Z
review text: Let u be the velocity of a fluid moving in a bounded region \(\Omega\) in \({\mathbb{R}}^ 3\). If u is regular enough, then a parametric representation of the trajectories of the particle of fluid which was at time \(t=0\) at point a of \(\Omega\) is the solution of the ordinary differential equation, \(d\phi (a,t)/dt=u(\phi (a,t),t),\quad t\in R_+, \Phi (a,0)=a.\) In this paper, we show that it is possible to define the trajectories \(\Phi\) of the fluid even if u is the weak solution of the Navier-Stokes equations that is known to exist for all times. We prove also that the mappings \(\Phi (.,t)\) are volume-preserving in some weak sense. These results are a consequence of an a priori estimate given by the authors in Comm. Part. Diff. Eq. 6, 329-359 (1981; Zbl 0472.35070).
Mathematics Subject Classification ID: 76D05
zbMATH DE Number: 3870822
zbMATH Keywords: bounded region
zbMATH Keywords: parametric representation of the trajectories of the particle of fluid
zbMATH Keywords: weak solution
zbMATH Keywords: volume-preserving
zbMATH Keywords: a priori estimate
author: Ciprian Foias
author: Colette Guillopé
author: R. Temam
MaRDI profile type: Publication
full work available at URL: 0022-0396(85)90065-8
OpenAlex ID: W2025156753
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description: scientific article; zbMATH DE number 3870822