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January  2014, 13(1): 331-346. doi: 10.3934/cpaa.2014.13.331

Vanishing viscosity limits for space-time periodic Hamilton-Jacobi equations

Eddaly Guerra 1, and  Héctor Sánchez-Morgado 1,

1. 

Instituto de Matemáticas, Universidad Nacional Autónoma de México, México DF 04510, Mexico, Mexico

Received  January 2013 Revised  May 2013 Published  July 2013

Extending previuos results ([16, 1, 7]), we study the vanishing viscosity limit of solutions of space-time periodic Hamilton-Jacobi-Belllman equations, assuming that the ``Aubry set'' is the union of a finite number of hyperbolic periodic orbits of the Hamiltonian flow.
Keywords: Aubry set., periodic Hamiltonian, Hamilton-Jacobi equation, Viscosity solution.
Mathematics Subject Classification: Primary: 37J50, 49L25; Secondary: 70H2.
Citation: Eddaly Guerra, Héctor Sánchez-Morgado. Vanishing viscosity limits for space-time periodic Hamilton-Jacobi equations. Communications on Pure & Applied Analysis, 2014, 13 (1) : 331-346. doi: 10.3934/cpaa.2014.13.331
References:
[1]

N. Anantharaman, R. Iturriaga, P. Padilla and H. Sánchez-Morgado, Physical solutions of the Hamilton-Jacobi equation, Disc. Cont. Dyn. Sys. Series B, 5 (2005), 513-528. doi: 10.3934/dcdsb.2005.5.513.  Google Scholar

[2]

M. Bardi, I. Capuzzo Dolcetta, "Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations,'' Birkhausser, 1997. Google Scholar

[3]

G. Barles, "Solutions de viscosité des équations de Hamilton Jacobi,'' Mathématiques et Applications 17, Springer-Verlag, 1994. Google Scholar

[4]

G. Barles and P. E. Souganidis, Space-time periodic solutions and long-time behavior of solutions to quasi-linear parabolic equations, SIAM J. Math. Anal., 32 (2001), 1311-1323. doi: 10.1137/S0036141000369344.  Google Scholar

[5]

P. Bernard, Smooth critical subsolutions of the Hamilton-Jacobi equation, Math. Res. Lett., 14 (2007), 503-511. Google Scholar

[6]

P. Bernard, Connecting orbits of time dependent Lagrangian systems, Ann. Inst. Fourier, Grenoble, 52 (2002), 1533-1568. doi: 10.5802/aif.1924.  Google Scholar

[7]

U. Bessi, Aubry-Mather theory and Hamilton-Jacobi equations, Comm. Math. Phys., 235 (2003), 495-511. doi: 10.1007/s00220-002-0781-5.  Google Scholar

[8]

G. Contreras and R. Iturriaga, Convex Hamiltonians without conjugate points, Erg. Th. Dynam. Sys., 19 (1999), 901-952. doi: 10.1017/S014338579913387X.  Google Scholar

[9]

G. Contreras, R. Iturriaga and H. Sánchez-Morgado, Weak solutions of the Hamilton Jacobi equation for Time Periodic Lagrangians,, \href{http://www.matem.unam.mx/hector/wham.pdf} {Preprint.}, ().   Google Scholar

[10]

M. G. Crandall, L. C. Evans and P.-L. Lions, Some properties of viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc., 282 (1984), 487-502. doi: 10.1090/S0002-9947-1984-0732102-X.  Google Scholar

[11]

L. C. Evans, "Partial Differential Equations,'' Graduate Studies in Mathematics 19, AMS, 1997. Google Scholar

[12]

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

[13]

A. Fathi, On existence of smooth critical subsolutions of the Hamilton-Jacobi equation, Pub. Mat. Uruguay, 12 (2011), 87-98. Google Scholar

[14]

W. Fleming and M. Soner, "Controlled Markov Processes and Viscosity Solutions,'' Springer 1993. Google Scholar

[15]

M. I. Freidlin and A. D. Wentzell, "Random Perturbations of Dynamical Systems,'' Springer 1998. Google Scholar

[16]

H. R. Jauslin, H. O. Kreiss and J. Moser, On the forced Burgers equation with periodic boundary conditions, "Differential Equations, La Pietra,'' Proc. of Symp. Pure Math., 65 (1996), 133-153. Google Scholar

[17]

D. Massart, Subsolution of time-periodic Hamilton-Jacobi equations, Erg. Th. Dynam. Sys., 27 (2007), 1253-1265. doi: 10.1017/S0143385707000089.  Google Scholar

[18]

T. Rockafellar, "Convex Analysis,'' Princeton University Press, 1972. Google Scholar

show all references

References:
[1]

N. Anantharaman, R. Iturriaga, P. Padilla and H. Sánchez-Morgado, Physical solutions of the Hamilton-Jacobi equation, Disc. Cont. Dyn. Sys. Series B, 5 (2005), 513-528. doi: 10.3934/dcdsb.2005.5.513.  Google Scholar

[2]

M. Bardi, I. Capuzzo Dolcetta, "Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations,'' Birkhausser, 1997. Google Scholar

[3]

G. Barles, "Solutions de viscosité des équations de Hamilton Jacobi,'' Mathématiques et Applications 17, Springer-Verlag, 1994. Google Scholar

[4]

G. Barles and P. E. Souganidis, Space-time periodic solutions and long-time behavior of solutions to quasi-linear parabolic equations, SIAM J. Math. Anal., 32 (2001), 1311-1323. doi: 10.1137/S0036141000369344.  Google Scholar

[5]

P. Bernard, Smooth critical subsolutions of the Hamilton-Jacobi equation, Math. Res. Lett., 14 (2007), 503-511. Google Scholar

[6]

P. Bernard, Connecting orbits of time dependent Lagrangian systems, Ann. Inst. Fourier, Grenoble, 52 (2002), 1533-1568. doi: 10.5802/aif.1924.  Google Scholar

[7]

U. Bessi, Aubry-Mather theory and Hamilton-Jacobi equations, Comm. Math. Phys., 235 (2003), 495-511. doi: 10.1007/s00220-002-0781-5.  Google Scholar

[8]

G. Contreras and R. Iturriaga, Convex Hamiltonians without conjugate points, Erg. Th. Dynam. Sys., 19 (1999), 901-952. doi: 10.1017/S014338579913387X.  Google Scholar

[9]

G. Contreras, R. Iturriaga and H. Sánchez-Morgado, Weak solutions of the Hamilton Jacobi equation for Time Periodic Lagrangians,, \href{http://www.matem.unam.mx/hector/wham.pdf} {Preprint.}, ().   Google Scholar

[10]

M. G. Crandall, L. C. Evans and P.-L. Lions, Some properties of viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc., 282 (1984), 487-502. doi: 10.1090/S0002-9947-1984-0732102-X.  Google Scholar

[11]

L. C. Evans, "Partial Differential Equations,'' Graduate Studies in Mathematics 19, AMS, 1997. Google Scholar

[12]

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

[13]

A. Fathi, On existence of smooth critical subsolutions of the Hamilton-Jacobi equation, Pub. Mat. Uruguay, 12 (2011), 87-98. Google Scholar

[14]

W. Fleming and M. Soner, "Controlled Markov Processes and Viscosity Solutions,'' Springer 1993. Google Scholar

[15]

M. I. Freidlin and A. D. Wentzell, "Random Perturbations of Dynamical Systems,'' Springer 1998. Google Scholar

[16]

H. R. Jauslin, H. O. Kreiss and J. Moser, On the forced Burgers equation with periodic boundary conditions, "Differential Equations, La Pietra,'' Proc. of Symp. Pure Math., 65 (1996), 133-153. Google Scholar

[17]

D. Massart, Subsolution of time-periodic Hamilton-Jacobi equations, Erg. Th. Dynam. Sys., 27 (2007), 1253-1265. doi: 10.1017/S0143385707000089.  Google Scholar

[18]

T. Rockafellar, "Convex Analysis,'' Princeton University Press, 1972. Google Scholar

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