A criterion for uniqueness of Lagrangian trajectories for weak solutions of the 3D Navier-Stokes equations
Robinson, James C. 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 0010-3616Full text not available from this repository.
Official URL: http://dx.doi.org/10.1007/s00220-009-0819-z
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|
|Official Date:||August 2009|
|Number of Pages:||8|
|Page Range:||pp. 15-22|
|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|
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