Lagrangian representation of a flow
Let u be the velocity of a fluid moving in a bounded region \(\Omega\) in \({\mathbb{R}}^ 3\). If u is regular enough, then a parametric representation of the trajectories of the particle of fluid which was at time \(t=0\) at point a of \(\Omega\) is the solution of the ordinary differential equation, \(d\phi (a,t)/dt=u(\phi (a,t),t),\quad t\in R_+, \Phi (a,0)=a.\) In this paper, we show that it is possible to define the trajectories \(\Phi\) of the fluid even if u is the weak solution of the Navier-Stokes equations that is known to exist for all times. We prove also that the mappings \(\Phi (.,t)\) are volume-preserving in some weak sense. These results are a consequence of an a priori estimate given by the authors in Comm. Part. Diff. Eq. 6, 329-359 (1981; Zbl 0472.35070).
parametric representation of the trajectories of the particle of fluid
scientific article; zbMATH DE number 3870822