-
Previous Article
Capacity drop and traffic control for a second order traffic model
- NHM Home
- This Issue
-
Next Article
A multiscale model reduction method for nonlinear monotone elliptic equations in heterogeneous media
The Lax-Oleinik semigroup on graphs
1. | CIMAT, A.P. 402 C.P. 3600, Guanajuato. Gto, México |
2. | Instituto de Matemáticas, Universidad Nacional Autónoma de México, Cd. de México C. P. 04510, México |
We consider Tonelli Lagrangians on a graph, define weak KAM solutions, which happen to be the fixed points of the Lax-Oleinik semi-group, and identify their uniqueness set as the Aubry set, giving a representation formula. Our main result is the long time convergence of the Lax Oleinik semi-group. It follows that weak KAM solutions are viscosity solutions of the Hamilton-Jacobi equation [
References:
[1] |
Y. Achdou, F. Camilli, A. Cutrí and N. Tchou,
Hamilton-Jacobi equations constrained on
networks, Nonlinear Differ. Equ. Appl., 20 (2013), 413-445.
doi: 10.1007/s00030-012-0158-1. |
[2] |
G. Barles, A. Briani and E. Chasseigne,
A Bellman approach for two-domains optimal control
problems in ${\mathbb{R}^n}$, ESAIM: Control, Optimisation and Calculus of Variations, 19 (2013), 710-739.
doi: 10.1051/cocv/2012030. |
[3] |
F. Camilli and D. Schieborn,
Viscosity solutions of Eikonal equations on topological networks, Calc. Var. Partial Differential Equatons, 46 (2013), 671-686.
doi: 10.1007/s00526-012-0498-z. |
[4] |
F. Camilli and C. Marchi,
A comparison among various notions of viscosity solution for
Hamilton-Jacobi equations on networks, J. Math. Anal. Appl., 407 (2013), 112-118.
doi: 10.1016/j.jmaa.2013.05.015. |
[5] |
A. Davini and A. Siconolfi,
A generalized dynamical approach to the large time behavior of
solutions of Hamilton-Jacobi equations, SIAM J. Math. Anal., 38 (2006), 478-502.
doi: 10.1137/050621955. |
[6] |
A. Fathi,
Sur la convergence du semi-groupe de Lax-Oleinik, C. R. Acad. Sci. Paris Sr. I Math., 327 (1998), 267-270.
doi: 10.1016/S0764-4442(98)80144-4. |
[7] |
A. Fathi,
Weak KAM Theorem in Lagrangian Dynamics, To appear in Cambridge Studies in Advanced Mathematics. |
[8] |
A. Fathi and A. Siconolfi,
PDE aspects of Aubry-Mather theory for quasi-convex Hamiltonians, Calc. Var. Partial Differential Equations, 22 (2005), 185-228.
doi: 10.1007/s00526-004-0271-z. |
[9] |
C. Imbert and R. Monneau,
Flux-limited Solutions for Quasi-Convex Hamilton-Jacobi Equations on Networks, arXiv: 1306.2428 |
[10] |
H. Ishii,
Asymptotic solutions for large time of Hamilton-Jacobi equations in Euclidean nspace, Anal. Non Linéaire, 25 (2008), 231-266.
doi: 10.1016/j.anihpc.2006.09.002. |
[11] |
J. M. Roquejoffre,
Convergence to steady states or periodic solutions in a class of HamiltonJacobi equations, J. Math. Pures Appl., 80 (2001), 85-104.
doi: 10.1016/S0021-7824(00)01183-1. |
show all references
References:
[1] |
Y. Achdou, F. Camilli, A. Cutrí and N. Tchou,
Hamilton-Jacobi equations constrained on
networks, Nonlinear Differ. Equ. Appl., 20 (2013), 413-445.
doi: 10.1007/s00030-012-0158-1. |
[2] |
G. Barles, A. Briani and E. Chasseigne,
A Bellman approach for two-domains optimal control
problems in ${\mathbb{R}^n}$, ESAIM: Control, Optimisation and Calculus of Variations, 19 (2013), 710-739.
doi: 10.1051/cocv/2012030. |
[3] |
F. Camilli and D. Schieborn,
Viscosity solutions of Eikonal equations on topological networks, Calc. Var. Partial Differential Equatons, 46 (2013), 671-686.
doi: 10.1007/s00526-012-0498-z. |
[4] |
F. Camilli and C. Marchi,
A comparison among various notions of viscosity solution for
Hamilton-Jacobi equations on networks, J. Math. Anal. Appl., 407 (2013), 112-118.
doi: 10.1016/j.jmaa.2013.05.015. |
[5] |
A. Davini and A. Siconolfi,
A generalized dynamical approach to the large time behavior of
solutions of Hamilton-Jacobi equations, SIAM J. Math. Anal., 38 (2006), 478-502.
doi: 10.1137/050621955. |
[6] |
A. Fathi,
Sur la convergence du semi-groupe de Lax-Oleinik, C. R. Acad. Sci. Paris Sr. I Math., 327 (1998), 267-270.
doi: 10.1016/S0764-4442(98)80144-4. |
[7] |
A. Fathi,
Weak KAM Theorem in Lagrangian Dynamics, To appear in Cambridge Studies in Advanced Mathematics. |
[8] |
A. Fathi and A. Siconolfi,
PDE aspects of Aubry-Mather theory for quasi-convex Hamiltonians, Calc. Var. Partial Differential Equations, 22 (2005), 185-228.
doi: 10.1007/s00526-004-0271-z. |
[9] |
C. Imbert and R. Monneau,
Flux-limited Solutions for Quasi-Convex Hamilton-Jacobi Equations on Networks, arXiv: 1306.2428 |
[10] |
H. Ishii,
Asymptotic solutions for large time of Hamilton-Jacobi equations in Euclidean nspace, Anal. Non Linéaire, 25 (2008), 231-266.
doi: 10.1016/j.anihpc.2006.09.002. |
[11] |
J. M. Roquejoffre,
Convergence to steady states or periodic solutions in a class of HamiltonJacobi equations, J. Math. Pures Appl., 80 (2001), 85-104.
doi: 10.1016/S0021-7824(00)01183-1. |
[1] |
Guangwu Wang, Boling Guo. Global weak solution to the quantum Navier-Stokes-Landau-Lifshitz equations with density-dependent viscosity. Discrete and Continuous Dynamical Systems - B, 2019, 24 (11) : 6141-6166. doi: 10.3934/dcdsb.2019133 |
[2] |
Mei Wang, Zilai Li, Zhenhua Guo. Global weak solution to 3D compressible flows with density-dependent viscosity and free boundary. Communications on Pure and Applied Analysis, 2017, 16 (1) : 1-24. doi: 10.3934/cpaa.2017001 |
[3] |
Vladimir E. Fedorov, Natalia D. Ivanova. Identification problem for a degenerate evolution equation with overdetermination on the solution semigroup kernel. Discrete and Continuous Dynamical Systems - S, 2016, 9 (3) : 687-696. doi: 10.3934/dcdss.2016022 |
[4] |
Cong Qin, Xinfu Chen. A new weak solution to an optimal stopping problem. Discrete and Continuous Dynamical Systems - B, 2020, 25 (12) : 4823-4837. doi: 10.3934/dcdsb.2020128 |
[5] |
Shuxing Chen, Jianzhong Min, Yongqian Zhang. Weak shock solution in supersonic flow past a wedge. Discrete and Continuous Dynamical Systems, 2009, 23 (1&2) : 115-132. doi: 10.3934/dcds.2009.23.115 |
[6] |
Diogo A. Gomes. Viscosity solution methods and the discrete Aubry-Mather problem. Discrete and Continuous Dynamical Systems, 2005, 13 (1) : 103-116. doi: 10.3934/dcds.2005.13.103 |
[7] |
Jutamas Kerdkaew, Rabian Wangkeeree. Characterizing robust weak sharp solution sets of convex optimization problems with uncertainty. Journal of Industrial and Management Optimization, 2020, 16 (6) : 2651-2673. doi: 10.3934/jimo.2019074 |
[8] |
Meng Wang, Wendong Wang, Zhifei Zhang. On the uniqueness of weak solution for the 2-D Ericksen--Leslie system. Discrete and Continuous Dynamical Systems - B, 2016, 21 (3) : 919-941. doi: 10.3934/dcdsb.2016.21.919 |
[9] |
Toyohiko Aiki, Adrian Muntean. On uniqueness of a weak solution of one-dimensional concrete carbonation problem. Discrete and Continuous Dynamical Systems, 2011, 29 (4) : 1345-1365. doi: 10.3934/dcds.2011.29.1345 |
[10] |
Yong Zeng. Existence and uniqueness of very weak solution of the MHD type system. Discrete and Continuous Dynamical Systems, 2020, 40 (10) : 5617-5638. doi: 10.3934/dcds.2020240 |
[11] |
Chérif Amrouche, María Ángeles Rodríguez-Bellido. On the very weak solution for the Oseen and Navier-Stokes equations. Discrete and Continuous Dynamical Systems - S, 2010, 3 (2) : 159-183. doi: 10.3934/dcdss.2010.3.159 |
[12] |
Keisuke Takasao. Existence of weak solution for mean curvature flow with transport term and forcing term. Communications on Pure and Applied Analysis, 2020, 19 (5) : 2655-2677. doi: 10.3934/cpaa.2020116 |
[13] |
Zhong Tan, Jianfeng Zhou. Higher integrability of weak solution of a nonlinear problem arising in the electrorheological fluids. Communications on Pure and Applied Analysis, 2016, 15 (4) : 1335-1350. doi: 10.3934/cpaa.2016.15.1335 |
[14] |
Hua Zhong, Chunlai Mu, Ke Lin. Global weak solution and boundedness in a three-dimensional competing chemotaxis. Discrete and Continuous Dynamical Systems, 2018, 38 (8) : 3875-3898. doi: 10.3934/dcds.2018168 |
[15] |
Alain Hertzog, Antoine Mondoloni. Existence of a weak solution for a quasilinear wave equation with boundary condition. Communications on Pure and Applied Analysis, 2002, 1 (2) : 191-219. doi: 10.3934/cpaa.2002.1.191 |
[16] |
Tong Li, Anthony Suen. Existence of intermediate weak solution to the equations of multi-dimensional chemotaxis systems. Discrete and Continuous Dynamical Systems, 2016, 36 (2) : 861-875. doi: 10.3934/dcds.2016.36.861 |
[17] |
Shijin Ding, Changyou Wang, Huanyao Wen. Weak solution to compressible hydrodynamic flow of liquid crystals in dimension one. Discrete and Continuous Dynamical Systems - B, 2011, 15 (2) : 357-371. doi: 10.3934/dcdsb.2011.15.357 |
[18] |
Mikhail Turbin, Anastasiia Ustiuzhaninova. Pullback attractors for weak solution to modified Kelvin-Voigt model. Evolution Equations and Control Theory, 2022 doi: 10.3934/eect.2022011 |
[19] |
Shanjian Tang, Fu Zhang. Path-dependent optimal stochastic control and viscosity solution of associated Bellman equations. Discrete and Continuous Dynamical Systems, 2015, 35 (11) : 5521-5553. doi: 10.3934/dcds.2015.35.5521 |
[20] |
J. Huang, Marius Paicu. Decay estimates of global solution to 2D incompressible Navier-Stokes equations with variable viscosity. Discrete and Continuous Dynamical Systems, 2014, 34 (11) : 4647-4669. doi: 10.3934/dcds.2014.34.4647 |
2021 Impact Factor: 1.41
Tools
Metrics
Other articles
by authors
[Back to Top]