Lagrangian and Eulerian specification of the flow field

Summary

NOTOC In classical field theories, the Lagrangian specification of the flow field is a way of looking at fluid motion where the observer follows an individual fluid parcel as it moves through space and time. Plotting the position of an individual parcel through time gives the pathline of the parcel. This can be visualized as sitting in a boat and drifting down a river. The Eulerian specification of the flow field is a way of looking at fluid motion that focuses on specific locations in the space through which the fluid flows as time passes. This can be visualized by sitting on the bank of a river and watching the water pass the fixed location. The Lagrangian and Eulerian specifications of the flow field are sometimes loosely denoted as the Lagrangian and Eulerian frame of reference. However, in general both the Lagrangian and Eulerian specification of the flow field can be applied in any observer's frame of reference, and in any coordinate

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On a class of mean field solutions of the Monge problem for perfect and self-interacting systems

Philippe Choquard

The Monge problem [23], [27], as reformulated by Kantorovich [19], [20] is that of the transportation, at a minimum "cost", of a given mass distribu- tion from an initial to a final position during a given time interval. It is an optimal transport problem [28, sects. 1, 2]. Following the fluid mechanical solution provided by Benamou and Brenier for quadratic cost functions [4] ,[28, sects. 5.4, 8.1] and, by analogy with the fixed end problem in Analytical Mechanics, Lagrangian formulations are needed to solve this boundary value problem in time. They are also needed to determine the Actions as time in- tegral of Lagrangians, that are measures of the "cost"of the transportations [4, proposition 1.1]. Four versions of explicit constructions of Lagrangians are proposed in section 3. They are associated to the Hamiltonians of perfect and self-interacting systems presented in section 2. These Hamiltonians are ex- pressed in function of pairs of the well known canonically conjugated Clebsch variables, namely mass densities and velocity potentials [14], [15]. The first version consists in the elimination of the velocity potentials as a function of the densities and their time derivatives by inversion of the continuity equations de- rived from given Hamiltonians. The second version consists in the elimination of the gradient of the velocity potentials from the continuity equations thanks to the introduction of vector valued applications such that their divergences give the mass densities. It turns out that, up to a sign factor, these vector elds are canonically conjugated to Euler velocity elds. The third version is a generalization in nD of Gelfand mass coordinate, a constant of the motion in 1 D [17], by the introduction of n-dimensional vector valued applications that enable to determine the mass densities as the determinant of their Jaco- bian matrices. Comparison of this set of mass coordinates with other sets of constants of the motion familiar in Fluid Dynamics is made in sub-section 3.3. Note that version two and three are identical for one-dimensional problems. The fourth version is based on the introduction of the Lagrangian coordinates that describe the characteristics of the different models and are parametrized by the former auxiliary vector fields. As illustrations, weak solutions of several models of Coulombian and Newtonian systems known in Plasma Physics and in Cosmology, respectively, with spherically symmetric boundary densities are given in section 4. However, and up to one exception given in the sub-section 3.3, calculations of the actions associated to these illustrations are not reported in this paper, nor the important analysis of the convexity-concavity properties of our Lagrangians. Lastly, and for the same models as those evoked above, a survey of past work concerning weak solutions of the Cauchy problem obeying the Hopf-Lax variational principle extended to negative time and having cor- related initial conditions is given in the Introduction as well as the derivation of the continuum fluid limit from many particle Hamiltonians.

EPFL2010

The Monge problem (Monge 1781; Taton 1951), as reformulated by Kantorovich (2006a, 2006b) is that of the transportation at a minimum "cost" of a given mass distribution from an initial to a final position during a given time interval. It is an optimal transport problem (Villani, 2003, sects. 1, 2). Following the fluid mechanical solution provided by Benamou and Brenier (2000) for quadratic cost functions (Villani, 2003, sects. 5.4, 8.1), Lagrangian formulations are needed to solve this boundary value problem in time and to determine the Actions as time integral of Lagrangians that are measures of the "cost" of the transportations (Benamou and Brenier, 2000, prop. 1.1). Given canonical Hamilltonians of perfect and self-interacting systems expressed in function of mass densities and velocity potentials, four versions of explicit constructions of Lagrangians, with their corresponding generalized coordinates, are proposed: elimination of the velocity potentials as a function of the densities and their time derivatives by inversion of the continuity equations; elimination of the gradient of the velocity potentials from the continuity equations thanks to the introduction of vector fields such that their divergences give the mass densities; generalization in nD of Gelfand mass coordinate (1963) by the introduction of n-dimensional vector fields such that the determinant of their Jacobian matrices give the mass densities; and, last, introduction of the Lagrangian coordinates that describe the characteristics of the different models and are parametrized by the former auxiliary vector fields. Using this version, weak solutions of several models of Coulombian and Newtonian systems known in Plasma Physics and in Cosmology, with spherically symmetric boundary densities, are given as illustrations.

2010

On a Lagrangian Formulation of the Incompressible Euler Equation

Hasan Inci

In this paper we show that the incompressible Euler equation on the Sobolev space H-s(R-n), s> n/2+1, can be expressed in Lagrangian coordinates as a geodesic equation on an infinite dimensional manifold. Moreover the Christoffel map describing the geodesic equation is real analytic. The dynamics in Lagrangian coordinates is described on the group of volume preserving diffeomorphisms, which is an analytic submanifold of the whole diffeomorphism group. Furthermore it is shown that a Sobolev class vector field integrates to a curve on the diffeomorphism group.

Global Science Press2016

On a class of mean field solutions of the Monge problem for perfect and self-interacting systems

Philippe Choquard

EPFL2010

Lagrangian and Eulerian specification of the flow field

On a class of mean fi eld solutions of the Monge problem for perfect and self-interacting systems

On a Lagrangian Formulation of the Incompressible Euler Equation

On a Lagrangian Formulation of the Incompressible Euler Equation

On a class of mean fi eld solutions of the Monge problem for perfect and self-interacting systems

On a class of mean field solutions of the Monge problem for perfect and self-interacting systems

On a class of mean field solutions of the Monge problem for perfect and self-interacting systems