Fathi, Albert; Mather, John N. Failure of convergence of the Lax-Oleinik semi-group in the time-periodic case. (English) Zbl 0989.37035 Bull. Soc. Math. Fr. 128, No. 3, 473-483 (2000). Summary: For a time-independent Lagrangian, the so-called Lax-Oleinik semigroup converges with an arbitrary continuous function as initial condition. Using twist maps, the authors show that there is no such convergence for time-periodic Lagrangians. Cited in 5 ReviewsCited in 16 Documents MSC: 37E40 Dynamical aspects of twist maps 35B40 Asymptotic behavior of solutions to PDEs 37J05 Relations of dynamical systems with symplectic geometry and topology (MSC2010) 35B10 Periodic solutions to PDEs 35F20 Nonlinear first-order PDEs Keywords:Lagrangian; Lax-Oleinik semigroup PDFBibTeX XMLCite \textit{A. Fathi} and \textit{J. N. Mather}, Bull. Soc. Math. Fr. 128, No. 3, 473--483 (2000; Zbl 0989.37035) Full Text: DOI Numdam EuDML References: [1] BANGERT (V.) . - Mather Sets for Twist Maps and Geodesics on Tori , Dyn. Rep., t. 1, 1988 , p. 1-56. MR 90a:58145 | Zbl 0664.53021 · Zbl 0664.53021 [2] BANGERT (V.) . - Geodesic Rays, Busemann Functions, and Monotone Twist Maps , Calc. Var., t. 2, 1994 , p. 49-63. MR 97b:53041 | Zbl 0794.58010 · Zbl 0794.58010 · doi:10.1007/BF01234315 [3] BARLES (G.) , SOUGANIDIS (P.E.) . - On the Large Time Behavior of Solutions of Hamilton-Jacobi Equations . - Preprint, 1998 . [4] FATHI (A.) . - Sur la convergence du semi-groupe de Lax-Oleinik , C. R. Acad. Sci. Paris, Série I, t. 327, 1998 , p. 267-270. MR 2000a:37058 | Zbl 01235022 · Zbl 1052.37514 [5] DENZLER (J.) . - Mather Sets for Plane Hamiltonian Systems , J. Applied Math. Phys. (ZAMP), t. 38, 1987 , p. 791-812. MR 89h:58055 | Zbl 0641.70014 · Zbl 0641.70014 · doi:10.1007/BF00945819 [6] MAà’É (R.) . - On the Minimizing Measures of Lagrangian Dynamical Systems , Nonlinearity, t. 5, 1992 , p. 623-638. MR 93h:58059 | Zbl 0799.58030 · Zbl 0799.58030 · doi:10.1088/0951-7715/5/3/001 [7] MATHER (J.) . - Differentiability of the Minimal Average Action as a Function of the Rotation Number , Bol. Soc. Bras. Mat., t. 21, 1990 , p. 59-70. MR 92j:58061 | Zbl 0766.58033 · Zbl 0766.58033 · doi:10.1007/BF01236280 [8] MATHER (J.) . - Action Minimizing Measures for Positive Definite Lagrangian Systems , Math. Z., t. 207, 1991 , p. 169-207. Article | MR 92m:58048 | Zbl 0696.58027 · Zbl 0696.58027 · doi:10.1007/BF02571383 [9] MATHER (J.) . - Variational Construction of Connecting Orbits , Ann. Inst. Fourier, Grenoble, t. 43, 1993 , p. 1349-1386. Numdam | MR 95c:58075 | Zbl 0803.58019 · Zbl 0803.58019 · doi:10.5802/aif.1377 [10] NAMAH, (G.) , ROQUEJOFFRE, (J.-M.) . - Comportement asymptotique des solutions d’une classe d’équations paraboliques et de Hamilton-Jacobi , C. R. Acad. Sci. Paris, Série I, t. 324, 1997 , p. 1367-1370. MR 98c:35078 | Zbl 0878.35056 · Zbl 0878.35056 · doi:10.1016/S0764-4442(97)83576-8 [11] NAMAH, (G.) , ROQUEJOFFRE, (J.-M.) . - Remarks on the Long Time Behavior of the Solutions of Hamilton-Jacobi Equations , Comm. Partial Diff. Eq., t. 5, 1999 , p. 883-894. MR 2000j:35034 | Zbl 0924.35028 · Zbl 0924.35028 · doi:10.1080/03605309908821451 [12] ROQUEJOFFRE (J.-M.) . - Comportement asymptotique des solutions d’équations de Hamilton-Jacobi monodimensionnelles , C. R. Acad. Sci. Paris, Série I, t. 326, 1998 , p. 185-189. MR 1646936 | Zbl 0924.70017 · Zbl 0924.70017 · doi:10.1016/S0764-4442(97)89468-2 [13] ROQUEJOFFRE (J.-M.) . - Convergence to Steady States or Periodic Solutions in a Class of Hamilton-Jacobi Equations . - Preprint, 1998 . This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.