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

Foias, Guillopé, & Temam showed in 1985 that for a given weak solution \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${u\in L^\infty(0,T;L^2)\cap L^2(0,T;H^1)}$$\end{document} of the three-dimensional Navier-Stokes equations on a domain Ω, one can define a ‘trajectory mapping’ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Phi:\Omega\times[0,T]\rightarrow\Omega}$$\end{document} that gives a consistent choice of trajectory through each initial condition \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${a\in\Omega,\,\xi_a(t)=\Phi(a,t)}$$\end{document} , 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.