Dashti, M. and Robinson, James C. (James Cooper), 1969-. (2009) A simple proof of uniqueness of the particle trajectories for solutions of the Navier-Stokes equations. Nonlinearity, Vol.22 (No.4). pp. 735-746. ISSN 0951-7715

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Abstract

We give a simple proof of the uniqueness of fluid particle trajectories corresponding to (1) the solution of the two-dimensional Navier-Stokes equations with an initial condition that is only square integrable and (2) the local strong solution of the three-dimensional equations with an H-1/2-regular initial condition, i. e. with the minimal Sobolev regularity known to guarantee uniqueness. This result was proved by Chemin and Lerner (1995 J. Diff. Eqns 121 314-28) using the Littlewood-Paley theory for the flow in the whole space R-d, d >= 2. We first show that the solutions of the differential equation. X = u(X, t) are unique if u is an element of L-p(0, T; H(d/2)-1) for some p > 1 and root t u is an element of L-2(0, T; H(d/2)+1). We then prove, using standard energy methods, that the solution of the Navier-Stokes equations with initial condition in H(d/2)-1 satisfies these conditions. This proof is also valid for the more physically relevant case of bounded domains.

Item Type:	Journal Article
Subjects:	Q Science > QA Mathematics Q Science > QC Physics
Divisions:	Faculty of Science > Mathematics
Journal or Publication Title:	Nonlinearity
Publisher:	Institute of Physics Publishing Ltd.
ISSN:	0951-7715
Official Date:	April 2009
Volume:	Vol.22
Number:	No.4
Number of Pages:	12
Page Range:	pp. 735-746
Identification Number:	10.1088/0951-7715/22/4/003
Status:	Peer Reviewed
Publication Status:	Published
Access rights to Published version:	Restricted or Subscription Access
Funder:	Royal Society (Great Britain), Leverhulme Trust (LT), Engineering and Physical Sciences Research Council (EPSRC), Royal Society University
Grant number:	ER/F050798/1 (EPSRC), EP/G007470/1 (EPSRC)
URI:	http://wrap.warwick.ac.uk/id/eprint/28324

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