Time-analyticity of Lagrangian particle trajectories in ideal fluid flow

Abstract

It is known that the Eulerian and Lagrangian structures of fluid flow can be drastically different; for example, ideal fluid flow can have a trivial (static) Eulerian structure, while displaying chaotic streamlines. Here we show that ideal flow with limited spatial smoothness (an initial vorticity that is just a little better than continuous), nevertheless has time-analytic Lagrangian trajectories before the initial limited smoothness is lost. For proving such results we use a little-known Lagrangian formulation of ideal fluid flow derived by Cauchy in 1815 in a manuscript submitted for a prize of the French Academy. This formulation leads to simple recurrence relations among the time-Taylor coefficients of the Lagrangian map from initial to current fluid particle positions; the coefficients can then be bounded using elementary methods. We first consider various classes of incompressible fluid flow, governed by the Euler equations, and then turn to highly compressible flow governed by the Euler-Poisson equations, a case of cosmological relevance. The recurrence relation associated to the Lagrangian formulation of these incompressible and the compressible problems are so closely related that the proofs of time-analyticity are basically identical.

Semiclassical path to cosmic large-scale structure

“…The considered flow is potential in Eulerian coordinates, which implies that the Eulerian curl of the Eulerian velocity vanishes, i.e., ∇ x × v = 0. By employing the Lagrangian map, the statement of zero vorticity translates in Lagrangian coordinates to the so-called Cauchy invariants, with components (i = 1, 2, 3) [12,36] ε ijk x l,jẋl,k = 0 ,…”

“…Furthermore, from theoretical grounds it is expected that such spurious effects could be enhanced at late times. Indeed, the convergence radius of LPT, which should set the maximal step-size for such algorithms, depends on the inverse of the norm of the velocity gradients [12,13]. As a consequence, at late times, when velocity gradients become large, the convergence radius will naturally shrink.…”

“…The central quantity is the Lagrangian displacement field that encodes how fluid elements are displaced as a function of time and initial (Lagrangian) position. The corresponding perturbative framework is usually called Lagrangian perturbation theory (LPT), and, although a challenge, allows to investigate the instance of shell-crossing by analytic means [12][13][14][15][16][17]. However, to update the gravitational potential that is responsible for displacing the fluid elements, one still requires, effectively, the Eulerian density.…”

Quasi-spherical collapse of matter in ΛCDM

“…We make use of the linear growth time a, which, for an EdS universe is identical to the cosmic scale factor. As pointed out by Zheligovsky & Frisch (2014); Rampf et al (2015); Rampf & Frisch (2017), enabling a as the time variable is essential when investigating the time-analyticity of the Lagrangian map. Before considering the Lagrangian-coordinates approach, let us briefly discuss the properties of the fluid equations at arbitrary short times.…”

Dynamics of The Tranquil Cosmic Web

“…, where the ZA term (s = 1) is entirely fixed by the initial conditions, while the EoM dictate all s > 1 terms via a recurrence relation involving lower order terms only (Zheligovsky & Frisch 2014). Unfortunately these recurrence relations become messy for s > 2.…”