American Institute of Mathematical Sciences

Advanced
December  2017, 12(4): 643-662. doi: 10.3934/nhm.2017026

The Lax-Oleinik semigroup on graphs

Renato Iturriaga 1, and  Héctor Sánchez Morgado 2,,

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

* Corresponding author:Morgado Héctor Sánchez

Received  June 2017 Revised  August 2017 Published  October 2017

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 [3, 4], and in the case of Hamiltonians called of eikonal type in [3], we prove that the converse holds.

Keywords: Lax-Oleinik semigroup, weak KAM solution, viscosity solution.
Mathematics Subject Classification: Primary: 35R02, 35F21; Secondary: 35Q93.
Citation: Renato Iturriaga, Héctor Sánchez Morgado. The Lax-Oleinik semigroup on graphs. Networks & Heterogeneous Media, 2017, 12 (4) : 643-662. doi: 10.3934/nhm.2017026
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

A. Fathi, Weak KAM Theorem in Lagrangian Dynamics, To appear in Cambridge Studies in Advanced Mathematics. Google Scholar

[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.  Google Scholar

[9]

C. Imbert and R. Monneau, Flux-limited Solutions for Quasi-Convex Hamilton-Jacobi Equations on Networks, arXiv: 1306.2428 Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

A. Fathi, Weak KAM Theorem in Lagrangian Dynamics, To appear in Cambridge Studies in Advanced Mathematics. Google Scholar

[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.  Google Scholar

[9]

C. Imbert and R. Monneau, Flux-limited Solutions for Quasi-Convex Hamilton-Jacobi Equations on Networks, arXiv: 1306.2428 Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[1]

Guangwu Wang, Boling Guo. Global weak solution to the quantum Navier-Stokes-Landau-Lifshitz equations with density-dependent viscosity. Discrete & 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 & 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 & 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 & 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 & 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 & 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 & 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 & 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 & 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 & 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 & 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 & 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 & 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 & 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 & 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 & 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 & Continuous Dynamical Systems - B, 2011, 15 (2) : 357-371. doi: 10.3934/dcdsb.2011.15.357
[18]

Andrea Davini, Maxime Zavidovique. Weak KAM theory for nonregular commuting Hamiltonians. Discrete & Continuous Dynamical Systems - B, 2013, 18 (1) : 57-94. doi: 10.3934/dcdsb.2013.18.57
[19]

Shanjian Tang, Fu Zhang. Path-dependent optimal stochastic control and viscosity solution of associated Bellman equations. Discrete & 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 & Continuous Dynamical Systems, 2014, 34 (11) : 4647-4669. doi: 10.3934/dcds.2014.34.4647

2020 Impact Factor: 1.213

Tools

Metrics

  • PDF downloads (57)
  • HTML views (229)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy