Almost-everywhere uniqueness of Lagrangian trajectories for suitable weak solutions of the three-dimensional Navier-Stokes equations
Robinson, James C. (James Cooper), 1969- and Sadowski, Witold. (2009) Almost-everywhere uniqueness of Lagrangian trajectories for suitable weak solutions of the three-dimensional Navier-Stokes equations. Nonlinearity, Vol.22 (No.9). pp. 2093-2099. ISSN 0951-7715Full text not available from this repository.
Official URL: http://dx.doi.org/10.1088/0951-7715/22/9/002
We show that if u is a suitable weak solution of the unforced three-dimensional Navier-Stokes equations corresponding to a divergence-free initial condition in H-1/2(Omega), then the Lagrangian trajectory through almost every initial condition in Omega is unique for all t >= 0. This is a corollary of two subsidiary results: (i) if S is the singular set of some suitable weak solution of the three-dimensional Navier-Stokes equations for which p is an element of L-5/3(Omega x (0, T)) (such solutions are known to exist), then for any compact subset K of Omega x (0, T) the upper box-counting dimension of S boolean AND K is no larger than 5/3 and (ii) for a volume-preserving flow in R-n arising from a vector field in L-1(0, T; L-infinity), trajectories through almost every initial condition avoid any set whose box-counting dimension is strictly less than n - 1. The result (i) is new, although it requires only a small variation of an argument contained in Caffarelli et al (1982 Commun. Pure Appl. Math. 35 771-831) and we give a simple geometric proof of (ii), a result originally proved by Aizenmann using other methods (1978 Duke Math. J. 45 809-12).
|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:||7|
|Page Range:||pp. 2093-2099|
|Access rights to Published version:||Restricted or Subscription Access|
|Funder:||Engineering and Physical Sciences Research Council (EPSRC), Polish Government|
|Grant number:||EP/G007470/1 (EPSRC), 1 P03A 017 30|
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