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arXiv:2410.20943 (Published 2024-10-28)
Long time behaviour of generalised gradient flows via occupational measures
This paper introduces new methods to study the long time behaviour of the generalised gradient flow associated with a solution of the critical equation for mechanical Hamiltonian system posed on the flat torus $\mathbb{T}^d$. For this analysis it is necessary to look at the critical set of $u$ consisting of all the points on $\mathbb{T}^d$ such that zero belongs to the super-differential of such a solution. Indeed, such a set turns out to be an attractor for the generalised gradient flow. Moreover, being the critical set the union of two subsets of rather different nature, namely the regular critical set and the singular set, we are interested in establishing whether the generalised gradient flow approaches the former or the latter as $t\to \infty$. One crucial tool of our analysis is provided by limiting occupational measures, a family of measures that are invariant under the generalized flow. Indeed, we show that by integrating the potential with respect to such measures, one can deduce whether the generalised gradient flow enters the singular set in finite time, or it approaches the regular critical set as time tends to infinity.
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arXiv:2008.05985 (Published 2020-08-13)
Local singular characteristics on $\mathbb{R}^2$
Comments: original research paper, 20 pages, 1 figureThe singular set of a viscosity solution to a Hamilton-Jacobi equation is known to propagate, from any noncritical singular point, along singular characteristics which are curves satisfying certain differential inclusions. In the literature, different notions of singular characteristics were introduced. However, a general uniqueness criterion for singular characteristics, not restricted to mechanical systems or problems in one space dimension, is missing at the moment. In this paper, we prove that, for a Tonelli Hamiltonian on $\mathbb{R}^2$, two different notions of singular characteristics coincide up to a bi-Lipschitz reparameterization. As a significant consequence, we obtain a uniqueness result for the class of singular characteristics that was introduced by Khanin and Sobolevski in the paper [On dynamics of Lagrangian trajectories for Hamilton-Jacobi equations. Arch. Ration. Mech. Anal., 219(2):861-885, 2016].
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arXiv:2004.06505 (Published 2020-04-14)
Weak KAM approach to first-order Mean Field Games with state constraints
We study the asymptotic behavior of solutions to the constrained MFG system as the time horizon $T$ goes to infinity. For this purpose, we analyze first Hamilton-Jacobi equations with state constraints from the viewpoint of weak KAM theory, constructing a Mather measure for the associated variational problem. Using these results, we show that a solution to the constrained ergodic mean field games system exists and the ergodic constant is unique. Finally, we prove that any solution of the first-order constrained MFG problem on $[0,T]$ converges to the solution of the ergodic system as $T \to +\infty$.
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arXiv:1912.04863 (Published 2019-12-10)
Singularities of solutions of time dependent Hamilton-Jacobi equations. Applications to Riemannian geometry
If $U:[0,+\infty[\times M$ is a uniformly continuous viscosity solution of the evolution Hamilton-Jacobi equation $$\partial_tU+ H(x,\partial_xU)=0,$$ where $M$ is a not necessarily compact manifold, and $H$ is a Tonelli Hamiltonian, we prove the set $\Sigma(U)$, of points where $U$ is not differentiable, is locally contractible. Moreover, we study the homotopy type of $\Sigma(U)$. We also give an application to the singularities of a distance function to a closed subset of a complete Riemannian manifold.
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arXiv:1809.09057 (Published 2018-09-24)
Long Time Behavior of First Order Mean Field Games on Euclidean Space
Categories: math.OCThe 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.
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arXiv:1605.07581 (Published 2016-05-24)
Generalized characteristics and Lax-Oleinik operators: global theory
For autonomous Tonelli systems on $\R^n$, we develop an intrinsic proof of the existence of generalized characteristics using sup-convolutions. This approach, together with convexity estimates for the fundamental solution, leads to new results such as the global propagation of singularities along generalized characteristics.