The Library
A criterion for uniqueness of Lagrangian trajectories for weak solutions of the 3D Navier-Stokes equations
Tools
Tools
Tools
Robinson, James C. (James Cooper), 1969- and Sadowski, Witold. (2009) A criterion for uniqueness of Lagrangian trajectories for weak solutions of the 3D Navier-Stokes equations. Communications in Mathematical Physics, Vol.290 (No.1). pp. 15-22. ISSN 0010-3616
Full text not available from this repository.
Official URL: http://dx.doi.org/10.1007/s00220-009-0819-z
Abstract
Foias, Guillope, & Temam showed in 1985 that for a given weak solution u is an element of L-infinity(0, T; L-2) boolean AND L-2(0, T; H-1) of the three-dimensional Navier-Stokes equations on a domain Omega, one can define a 'trajectory mapping' Phi: Omega x [0, T] -> Omega that gives a consistent choice of trajectory through each initial condition a is an element of Omega, xi a(t) = Phi(a, t), and that respects the volume-preserving property one would expect for smooth flows. The uniqueness of this mapping is guaranteed by the theory of renormalised solutions of non-smooth ODEs due to DiPerna & Lions. However, this is a distinct question from the uniqueness of individual particle trajectories. We show here that if one assumes a little more regularity for u than is known to be the case, namely that u is an element of L-6/5(0, T; L-infinity(Omega)), then the particle trajectories are unique and C-1 in time for almost every choice of initial condition in Omega. This degree of regularity is more than can currently be guaranteed for weak solutions (u is an element of L-1(0, T; L-infinity)) but significantly less than that known to ensure that u is regular (u is an element of L-2(0, T; L-infinity)). We rely heavily on partial regularity results due to Caffarelli, Kohn, & Nirenberg and Ladyzhenskaya & Seregin.
| Item Type: | Journal Article |
|---|---|
| Subjects: | Q Science > QC Physics |
| Divisions: | Faculty of Science > Mathematics |
| Journal or Publication Title: | Communications in Mathematical Physics |
| Publisher: | Springer |
| ISSN: | 0010-3616 |
| Date: | August 2009 |
| Volume: | Vol.290 |
| Number: | No.1 |
| Number of Pages: | 8 |
| Page Range: | pp. 15-22 |
| Identification Number: | 10.1007/s00220-009-0819-z |
| Status: | Peer Reviewed |
| Publication Status: | Published |
| Access rights to Published version: | Restricted or Subscription Access |
| Funder: | Leverhulme Trust (LT), Engineering and Physical Sciences Research Council (EPSRC), Polish Government |
| Grant number: | EP/G007470/1 (EPSRC), 1 P03A 017 30 |
| URI: | http://wrap.warwick.ac.uk/id/eprint/27702 |
Data sourced from Thomson Reuters' Web of Knowledge
Actions (login required)
 |
View Item |
