Uriel Frisch

Solving linear stochastic differential equations

The aim of this paper is to provide the user with tools for the solution of linear differential equations with random coefficients. Only analytic methods which lead to expressions in closed form for first and second order moments and probability distributions of the solution are considered. The paper deals both with approximate methods which require the existence of a small (or large) dimensionless parameter and with the method of model coefficients, where the true coefficients of the stochastic equation are replaced by random step functions with the same first and second order moments and probability distributions, chosen in such a way that the equation can be solved analytically. The second procedure does not rely on the existence of a small parameter.

Time-analyticity of Lagrangian particle trajectories in ideal fluid flow

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. To prove these 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 relations associated with the Lagrangian formulation of these incompressible and compressible problems are so closely related that the proofs of time-analyticity are basically identical.

Cauchy’s almost forgotten Lagrangian formulation of the Euler equation for 3D incompressible flow

AbstractnTwo prized papers, one by Augustin Cauchy in 1815, presented to the French Academy andnthe other by Hermann Hankel in 1861, presented to Göttingen University, contain majorndiscoveries on vorticity dynamics whose impact is now quickly increasing. Cauchy found anLagrangian formulation of 3D ideal incompressible flow in terms of three invariants thatngeneralize to three dimensions the now well-known law of conservation of vorticity alongnfluid particle trajectories for two-dimensional flow. This has very recently been used tonprove analyticity in time of fluid particle trajectories for 3D incompressible Euler flownand can be extended to compressible flow, in particular to cosmological dark matter.nHankel showed that Cauchy’s formulation gives a very simple Lagrangian derivation of thenHelmholtz vorticity-flux invariants and, in the middle of the proof, derived annintermediate result which is the conservation of the circulation of the velocity around anclosed contour moving with the fluid. This circulation theorem was to be rediscoverednindependently by William Thomson (Kelvin) in 1869. Cauchy’s invariants were onlynoccasionally cited in the 19th century – besides Hankel, foremost by George Stokes andnMaurice Lévy – and even less so in the 20th until they were rediscovered via EmmynNoether’s theorem in the late 1960, but reattributed to Cauchy only at the end of the 20thncentury by Russian scientists.n

The Monge-Ampère equation: Various forms and numerical solution

We present three novel forms of the Monge-Ampere equation, which is used, e.g., in image processing and in reconstruction of mass transportation in the primordial Universe. The central role in this paper is played by our Fourier integral form, for which we establish positivity and sharp bound properties of the kernels. This is the basis for the development of a new method for solving numerically the space-periodic Monge-Ampere problem in an odd-dimensional space. Convergence is illustrated for a test problem of cosmological type, in which a Gaussian distribution of matter is assumed in each localised object, and the right-hand side of the Monge-Ampere equation is a sum of such distributions.

A Very Smooth Ride in a Rough Sea

It has been known for some time that a 3D incompressible Euler flow that has initially a barely smooth velocity field nonetheless has Lagrangian fluid particle trajectories that are analytic in time for at least a finite time Serfati (C. R. Acad. Sci. Paris Série I 320:175–180, 1995), Shnirelman (Glob. Stoch. Anal., http://arxiv.org/abs/1205.5837v1, 2012). Here an elementary derivation is given, based on Cauchy’s form of the Euler equations in Lagrangian coordinates. This form implies simple recurrence relations among the time-Taylor coefficients of the Lagrangian map, used here to derive bounds for the C1,γ Hölder norms of the coefficients and infer temporal analyticity of Lagrangian trajectories when the initial velocity is C1,γ.

How smooth are particle trajectories in a Lambda CDM Universe

It is shown here that in a flat, cold dark matter (CDM) dominated Universe with positive cosmological constant (

Geometric formulation of the Cauchy invariants for incompressible Euler flow in flat and curved spaces

Lambda

The Cauchy-Lagrangian method for numerical analysis of Euler flow

), modelled in terms of a Newtonian and collisionless fluid, particle trajectories are analytical in time (representable by a convergent Taylor series) until at least a finite time after decoupling. The time variable used for this statement is the cosmic scale factor, i.e., the

Shell-crossing in quasi-one-dimensional flow

a

A Constructive Approach to Regularity of Lagrangian Trajectories for Incompressible Euler Flow in a Bounded Domain

-time, and not the cosmic time. For this, a Lagrangian-coordinates formulation of the Euler-Poisson equations is employed, originally used by Cauchy for 3-D incompressible flow. Temporal analyticity for