On the regularity of Lagrangian trajectories corresponding to suitable weak solutions of the Navier-Stokes equations

Article


Robinson, J., Sadowski, W. and Sharples, N. 2013. On the regularity of Lagrangian trajectories corresponding to suitable weak solutions of the Navier-Stokes equations. Procedia IUTAM. 7, pp. 161-166. https://doi.org/10.1016/j.piutam.2013.03.019
TypeArticle
TitleOn the regularity of Lagrangian trajectories corresponding to suitable weak solutions of the Navier-Stokes equations
AuthorsRobinson, J., Sadowski, W. and Sharples, N.
Abstract

The putative singular set S in space-time of a suitable weak solution u of the 3D Navier–Stokes equations has box-counting dimension no greater than 5/3. This allows one to prove that almost all trajectories avoid S. Moreover, for each point x that does not belong to S, one can find a neighbourhood U of x such that the function u is continuous on U and space derivatives of u are bounded on every compact subset of U. It follows that almost all Lagrangian trajectories corresponding to u are C^{1} functions of time (Robinson & Sadowski, Nonlinearity 2009). We recall the main idea of the proof, give examples that clarify in what sense the uniqueness of trajectories is considered, and make some comments on how this result might be improved.

LanguageEnglish
PublisherElsevier
JournalProcedia IUTAM
ISSN2210-9838
Publication dates
Print31 Dec 2013
Publication process dates
Deposited15 Oct 2015
Accepted08 Apr 2013
Output statusPublished
Publisher's version
Accepted author manuscript
File Access Level
Restricted
Copyright Statement

Published version: © 2013 The Authors. Published by Elsevier B.V. Open access under CC BY-NC-ND license -http://creativecommons.org/licenses/by-nc-nd/3.0/

Web address (URL)http://www.sciencedirect.com/science/article/pii/S2210983813000436
Digital Object Identifier (DOI)https://doi.org/10.1016/j.piutam.2013.03.019
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