Lagrangian dynamics on an infinite-dimensional torus; a Weak KAM theorem

Gangbo, Wilfrid; Tudorascu, Adrian

doi:10.1016/j.aim.2009.11.005

Cited by 20 publications

(54 citation statements)

References 12 publications

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“…An idea underlying several papers (see for instance [1], [9], [10], [12]) is to consider the Vlasov equation as a Hamiltonian system with infinitely many particles, i. e. as a Hamiltonian system on the space M 1 (T p ) of probability measures on T p ; in particular, one can define, on M 1 (T p ), both the Hopf-Lax semigroup and the Hamilton-Jacobi equation.…”

Section: Point Theory and Perturbative Methods For Nonlinear Differenmentioning

confidence: 99%

A time-step approximation scheme for a viscous version of the Vlasov equation

Bessi

2014

Advances in Mathematics

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Gomes and Valdinoci have introduced a time-step approximation scheme for a viscous version of AubryMather theory; this scheme is a variant of that of Jordan, Kinderlehrer and Otto. Gangbo and Tudorascu have shown that the Vlasov equation can be seen as an extension of Aubry-Mather theory, in which the configuration space is the space of probability measures, i. e. the different distributions of infinitely many particles on a manifold. Putting the two things together, we show that Gomes and Valdinoci's theorem carries over to a viscous version of the Vlasov equation. In this way, we shall recover a theorem of J. Feng and T. Nguyen, but by a different and more "elementary" proof.

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Section: Point Theory and Perturbative Methods For Nonlinear Differenmentioning

confidence: 99%

A time-step approximation scheme for a viscous version of the Vlasov equation

Bessi

2014

Advances in Mathematics

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“…Subsequent work [11,[13][14][15] upheld this restriction for the most part, although some interesting homogenization results were proved in the general case in [15]. Subsequent work [11,[13][14][15] upheld this restriction for the most part, although some interesting homogenization results were proved in the general case in [15].…”

Section: Introductionmentioning

confidence: 99%

“…In [13,14] the manifold considered is an appropriate factorization of the set of monotone nondecreasing L 2 .0; 1/-functions. This is isometric with the space P.T / (Borel probabilities on the one-dimensional torus), which is compact in the topology induced by the Z-periodic 2-Wasserstein distance (details in [13] and below).…”

Section: Introductionmentioning

confidence: 99%

“…a for all a 2 R 2d . General Hamiltonians such as (1.2) have been considered in [11][12][13][14][15], with applications ranging from action-minimizing solutions for the Euler-Poisson system to globally minimizing trajectories of prescribed asymptotic behavior for the nonlinear Vlasov or Vlasov-Poisson on the torus. When J is the block matrix J D .0; id d ; 0; id d / we get kinetic systems such as the linear or nonlinear Vlasov, and Vlasov-Monge-Ampère.…”

Section: Introductionmentioning

confidence: 99%

“…As far as the restriction to the one-dimensional setting is concerned, the main reason was an apparent lack of necessary compactness in general, which, solely in the d D 1 case, was offset by the isometric identification of the 2-Wasserstein space with the convex cone of nondecreasing L 2 .0; 1/-functions. Subsequent work [11,[13][14][15] upheld this restriction for the most part, although some interesting homogenization results were proved in the general case in [15]. The authors have searched for the correct setting and methodology to extend infinite-dimensional weak KAM results from the P 2 .R/ case developed in [13,14] to the P 2 .R d / case ever since those works were in progress.…”

Section: Introductionmentioning

confidence: 99%

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Weak KAM Theory on the Wasserstein Torus with Multidimensional Underlying Space

Gangbo

Tudorascu

2013

Comm Pure Appl Math

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The study of asymptotic behavior of minimizing trajectories on the Wasserstein space P.T d / has so far been limited to the case d D 1 as all prior studies heavily relied on the isometric identification of P.T / with a subset of the Hilbert space L 2 .0; 1/. There is no known analogue isometric identification when d > 1. In this article we propose a new approach, intrinsic to the Wasserstein space, which allows us to prove a weak KAM theorem on P.T d /, the space of probability measures on the torus, for any d 1. This space is analyzed in detail, facilitating the study of the asymptotic behavior/invariant measures associated with minimizing trajectories of a class of Lagrangians of practical importance.

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The Aubry set for a version of the Vlasov equation

Bessi

2013

Nonlinear Differ. Equ. Appl.

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Abstract.We check that several properties of the Aubry set, first proven for finite-dimensional Lagrangians by Mather and Fathi, continue to hold in the case of the infinitely many interacting particles of the Vlasov equation on the circle. Mathematics Subject Classification. Primary 37J50; Secondary 35Q83. Keywords. Vlasov equation, Aubry-Mather theory. IntroductionThe Vlasov equation on the circle governs the motion of a group of particles on S 1 := R Z under the action of an external potential V (t, x) and a mutual interaction W . More precisely, we let I = [0, 1), we lift our particles to R, and we parametrize them at time t by a function σ t ∈ L 2 (I, R); we require that σ t satisfies the differential equation in L 2 (I, R)Our standing hypotheses on the potentials V and W are; moreover W , seen as a function on R, is even; up to adding a constant, we can suppose that W (0) = 0.There is an element of arbitrariness in choosing the lift of the particles to R and in parametrizing them; that's why we are less interested in the evolution of the labelling σ t than in the evolution of the measure it induces. In other words, we want to study the measures on S 1 × R given by μ t := (π • σ t ,σ t ) ν 0 , where ν 0 denotes the Lebesgue measure on I, π : R → S 1 is the natural projection and (·) denotes the push-forward. A standard calculation shows that, if σ t satisfies (ODE) Lag , then μ t satisfies, in the weak sense,

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Lagrangian dynamics on an infinite-dimensional torus; a Weak KAM theorem

Cited by 20 publications

References 12 publications

A time-step approximation scheme for a viscous version of the Vlasov equation

A time-step approximation scheme for a viscous version of the Vlasov equation

Weak KAM Theory on the Wasserstein Torus with Multidimensional Underlying Space

The Aubry set for a version of the Vlasov equation

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