Failure of convergence of the Lax-Oleinik semi-group in the time periodic case

DOI10.24033/bsmf.2378zbMath0989.37035OpenAlexW135711142MaRDI QIDQ4509124

Albert Fathi, John N. Mather

Publication date: 11 October 2000

Published in: Bulletin de la Société mathématique de France (Search for Journal in Brave)

Full work available at URL: http://www.numdam.org/item?id=BSMF_2000__128_3_473_0

zbMATH Keywords

LagrangianLax-Oleinik semigroup

Mathematics Subject Classification ID

Asymptotic behavior of solutions to PDEs (35B40) Periodic solutions to PDEs (35B10) Nonlinear first-order PDEs (35F20) Dynamical aspects of twist maps (37E40)

Related Items (16)

Large time behavior of solutions for a class of time-dependent Hamilton-Jacobi equations ⋮ A necessary and sufficient condition for convergence of the Lax-Oleinik semigroup for reversible Hamiltonians on \(\mathbf{R}^{n}\) ⋮ Weak KAM theory for Hamilton-Jacobi equations depending on unknown functions ⋮ Time periodic solutions of Hamilton-Jacobi equations with autonomous Hamiltonian on the circle ⋮ The convergence of Lax-Oleinik semigroup for time-periodic Lagrangian ⋮ The asymptotic bounds of viscosity solutions of the Cauchy problem for Hamilton-Jacobi equations ⋮ Mather theory for piecewise smooth Lagrangian systems ⋮ On commuting Tonelli Hamiltonians: time-periodic case ⋮ A new kind of Lax-Oleinik type operator with parameters for time-periodic positive definite Lagrangian systems ⋮ Convergence of the viscosity solution of non-autonomous Hamilton-Jacobi equations ⋮ An example of a nearly integrable Hamiltonian system with a trajectory dense in a set of maximal Hausdorff dimension ⋮ Some Results on Weak KAM Theory for Time-periodic Tonelli Lagrangian Systems ⋮ Weak KAM theory for action minimizing random walks ⋮ Asymptotic solution and effective Hamiltonian of a Hamilton-Jacobi equation in the modeling of traffic flow on a homogeneous signalized road ⋮ Connecting orbits of time dependent Lagrangian systems ⋮ Lipschitz dependence of viscosity solutions of Hamilton-Jacobi equations with respect to the parameter

Cites Work

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• Action minimizing invariant measures for positive definite Lagrangian systems
• Mather sets for plane Hamiltonian systems
• Geodesic rays, Busemann functions and monotone twist maps
• Variational construction of connecting orbits
• Differentiability of the minimal average action as a function of the rotation number
• On the minimizing measures of Lagrangian dynamical systems
• Comportement asymptotique des solutions d'équations de Hamilton-Jacobi monodimensionnelles
• Sur la convergence du semi-groupe de Lax-Oleinik
• Remarks on the long time behaviour of the solutions of hamilton-jacobi equations
• Comportement asymptotique des solutions d'une classe d'équations paraboliques et de Hamilton-Jacobi

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