Weak KAM Theorem Research Articles

Long-Time Behavior of First-Order Mean Field Games on Euclidean Space

The aim of this paper is to study the long time behavior of solutions to deterministic mean field games systems on Euclidean space. This problem was addressed on the torus ${\mathbb T}^n$ in [P. Cardaliaguet, {\it Long time average of first order mean field games and weak KAM theory}, Dyn. Games Appl. 3 (2013), 473-488], where solutions are shown to converge to the solution of a certain ergodic mean field games system on ${\mathbb T}^n$. By adapting the approach in [A. Fathi, E. Maderna, {\it Weak KAM theorem on non compact manifolds}, NoDEA Nonlinear Differential Equations Appl. 14 (2007), 1-27], we identify structural conditions on the Lagrangian, under which the corresponding ergodic system can be solved in $\mathbb{R}^{n}$. Then we show that time dependent solutions converge to the solution of such a stationary system on all compact subsets of the whole space.

Weak KAM theory for a weakly coupled system of Hamilton–Jacobi equations

Here, we extend the weak KAM and Aubry–Mather theories to optimal switching problems. We consider three issues: the analysis of the calculus of variations problem, the study of a generalized weak KAM theorem for solutions of weakly coupled systems of Hamilton–Jacobi equations, and the long-time behavior of time-dependent systems. We prove the existence and regularity of action minimizers, obtain necessary conditions for minimality, extend Fathi’s weak KAM theorem, and describe the asymptotic limit of the generalized Lax–Oleinik semigroup.

Fast weak–KAM integrators for separable Hamiltonian systems

We consider a numerical scheme for Hamilton-Jacobi equations based on a direct discretization of the Lax-Oleinik semi-group. We prove that this method is convergent with respect to the time and space stepsizes provided the solution is Lipschitz, and give an error estimate. Moreover, we prove that the numerical scheme is a {\em geometric integrator} satisfying a discrete weak-KAM theorem which allows to control its long time behavior. Taking advantage of a fast algorithm for computing min-plus convolutions based on the decomposition of the function into concave and convex parts, we show that the numerical scheme can be implemented in a very efficient way.

On the free time minimizers of the NewtonianN-body problem

AbstractIn this paper we study the existence and the dynamics of a very special class of motions, which satisfy a strong global minimization property. More precisely, we call a free time minimizer a curve which satisfies the least action principle between any pair of its points without the constraint of time for the variations. An example of a free time minimizer defined on an unbounded interval is a parabolic homothetic motion by a minimal central configuration. The existence of a large amount of free time minimizers can be deduced from the weak KAM theorem. In particular, for any choice ofx0, there should be at least one free time minimizerx(t)defined for allt≥ 0 and satisfyingx(0)=x0. We prove that such motions are completely parabolic. Using Marchal's theorem we deduce as a corollary that there are no entire free time minimizers, i.e. defined on$\mathbb{R}$. This means that the Mañé set of the NewtonianN-body problem is empty.

Weak KAM Theory on the Wasserstein Torus with Multidimensional Underlying Space

The study of asymptotic behavior of minimizing trajectories on the Wasserstein space ��(��d) has so far been limited to the case d = 1 as all prior studies heavily relied on the isometric identification of ��(��) with a subset of the Hilbert space L2(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 ��(��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. © 2014 Wiley Periodicals, Inc.

The Lax–Oleinik semi-group: a Hamiltonian point of view

The weak KAM theory was developed by Fathi in order to study the dynamics of convex Hamiltonian systems. It somehow makes a bridge between viscosity solutions of the Hamilton–Jacobi equation and Mather invariant sets of Hamiltonian systems, although this was fully understood only a posteriori. These theories converge under the hypothesis of convexity, and the richness of applications mostly comes from this remarkable convergence. In this paper, we provide an elementary exposition of some of the basic concepts of weak KAM theory. In a companion paper, Albert Fathi exposed the aspects of his theory which are more directly related to viscosity solutions. Here, on the contrary, we focus on dynamical applications, even if we also discuss some viscosity aspects to underline the connections with Fathi's lecture. The fundamental reference on weak KAM theory is the still unpublished book Weak KAM theorem in Lagrangian dynamics by Albert Fathi. Although we do not offer new results, our exposition is original in several aspects. We only work with the Hamiltonian and do not rely on the Lagrangian, even if some proofs are directly inspired by the classical Lagrangian proofs. This approach is made easier by the choice of a somewhat specific setting. We work on ℝd and make uniform hypotheses on the Hamiltonian. This allows us to replace some compactness arguments by explicit estimates. For the most interesting dynamical applications, however, the compactness of the configuration space remains a useful hypothesis and we retrieve it by considering periodic (in space) Hamiltonians. Our exposition is centred on the Cauchy problem for the Hamilton–Jacobi equation and the Lax–Oleinik evolution operators associated to it. Dynamical applications are reached by considering fixed points of these evolution operators, the weak KAM solutions. The evolution operators can also be used for their regularizing properties; this opens an alternative route to dynamical applications.

On weak KAM theory for N-body problems

AbstractWe consider N-body problems with potential 1/r2κ, where κ∈(0,1), including the Newtonian case (κ=1/2). Given R>0 and T>0, we find a uniform upper bound for the minimal action of paths binding, in time T, any two configurations which are contained in some ball of radius R. Using cluster partitions, we obtain from these estimates the Hölder regularity of the critical action potential (i.e. of the minimal action of paths binding two configurations in free time). As an application, we establish the weak KAM theorem for these N-body problems, i.e. we prove the existence of fixed points of the Lax–Oleinik semigroup, and we show that they are global viscosity solutions of the corresponding Hamilton–Jacobi equation. We also prove that there are invariant solutions for the action of isometries on the configuration space.

Continuity of optimal control costs and its application to weak KAM theory

We prove continuity of certain cost functions arising from optimal control of affine control systems. We give sharp sufficient conditions for this continuity. As an application, we prove a version of weak KAM theorem and consider the Aubry–Mather problems corresponding to these systems.

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

The space L 2 ( 0 , 1 ) has a natural Riemannian structure on the basis of which we introduce an L 2 ( 0 , 1 ) -infinite-dimensional torus T . For a class of Hamiltonians defined on its cotangent bundle we establish existence of a viscosity solution for the cell problem on T or, equivalently, we prove a Weak KAM theorem. As an application, we obtain existence of absolute action-minimizing solutions of prescribed rotation number for the one-dimensional nonlinear Vlasov system with periodic potential.

Weak kam theorem on non compact manifolds

In this paper, we consider a time independent C2 Hamiltonian, satisfying the usual hypothesis of the classical Calculus of Variations, on a non-compact connected manifold. Using the Lax-Oleinik semigroup, we give a proof of the existence of weak KAM solutions, or viscosity solutions, for the associated Hamilton-Jacobi Equation. This proof works also in presence of symmetries. We also study the role of the amenability of the group of symmetries to understand when the several critical values that can be associated with the Hamiltonian coincide.

Solutions KAM faibles conjuguées et barrières de Peierls

We continue our study of the weak KAM theorem, which we obtained in our previous Note. We state the connection between this theorem and the Peierls's barriers as defined by Mather. From this connection one can deduce the fact, due to Mañé, that the dynamics on the set of M-minimizing extremals i.s chain transitive.