A simple proof of uniqueness of the particle trajectories for solutions of the Navier-Stokes equations
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 0951-7715Full text not available from this repository.
Official URL: http://dx.doi.org/10.1088/0951-7715/22/4/003
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.|
|Number of Pages:||12|
|Page Range:||pp. 735-746|
|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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