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A simple proof of uniqueness of the particle trajectories for solutions of the Navier-Stokes equations
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Dashti, M. and Robinson, James C. (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. doi:10.1088/0951-7715/22/4/003
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Official URL: http://dx.doi.org/10.1088/0951-7715/22/4/003
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 | ||||
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| Subjects: | Q Science > QA Mathematics Q Science > QC Physics |
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| Divisions: | Faculty of Science > Mathematics | ||||
| Journal or Publication Title: | Nonlinearity | ||||
| Publisher: | Institute of Physics Publishing Ltd. | ||||
| ISSN: | 0951-7715 | ||||
| Official Date: | April 2009 | ||||
| Dates: |
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| Volume: | Vol.22 | ||||
| Number: | No.4 | ||||
| Number of Pages: | 12 | ||||
| Page Range: | pp. 735-746 | ||||
| DOI: | 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) |
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