A Criterion for Uniqueness of Lagrangian Trajectories for Weak Solutions of the 3D Navier-Stokes Equations期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

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A Criterion for Uniqueness of Lagrangian Trajectories for Weak Solutions of the 3D Navier-Stokes Equations

Authors:	James C Robinson and Witold Sadowski

Institution:	(1) Mathematics Institute, University of Warwick, Coventry, CV4 7AL, UK;(2) Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Banacha 2, 02-097 Warszawa, Poland

Abstract:	Foias, Guillopé, & Temam showed in 1985 that for a given weak solution ${u\in L^\infty(0,T;L^2)\cap L^2(0,T;H^1)}$ of the three-dimensional Navier-Stokes equations on a domain Ω, one can define a ‘trajectory mapping’ ${\Phi:\Omega\times0,T]\rightarrow\Omega}$ that gives a consistent choice of trajectory through each initial condition ${a\in\Omega,\,\xi_a(t)=\Phi(a,t)}$ , 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 ${u\in L^{6/5}(0,T;L^\infty(\Omega))}$ , then the particle trajectories are unique and C ¹ in time for almost every choice of initial condition in Ω. This degree of regularity is more than can currently be guaranteed for weak solutions ( ${u\in L^1(0,T;L^\infty)}$ ) but significantly less than that known to ensure that u is regular ( ${u\in L^2(0,T;L^\infty))}$ . We rely heavily on partial regularity results due to Caffarelli, Kohn, & Nirenberg and Ladyzhenskaya & Seregin.

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