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Averaging of Hamilton-Jacobi equations along divergence-free vector fields
1. | Institute for Mathematics and Computer Science, Tsuda University, 2-1-1 Tsuda, Kodaira, Tokyo 187-8577 Japan |
2. | National Institute of Technology, Maizuru College, 234 Shiroya, Maizuru-shi, Kyoto 625-8511 Japan |
We study the asymptotic behavior of solutions to the Dirichlet problem for Hamilton-Jacobi equations with large drift terms, where the drift terms are given by divergence-free vector fields. This is an attempt to understand the averaging effect for fully nonlinear degenerate elliptic equations. In this work, we restrict ourselves to the case of Hamilton-Jacobi equations. The second author has already established averaging results for Hamilton-Jacobi equations with convex Hamiltonians ($ G $ below) under the classical formulation of the Dirichlet condition. Here we treat the Dirichlet condition in the viscosity sense and establish an averaging result for Hamilton-Jacobi equations with relatively general Hamiltonian $ G $.
References:
[1] |
Y. Achdou and N. Tchou,
Hamilton-Jacobi equations on networks as limits of singularly perturbed problems in optimal control: dimension reduction, Comm. Partial Differential Equations, 40 (2015), 652-693.
doi: 10.1080/03605302.2014.974764. |
[2] |
Y. Achdou, F. Camilli, A. Cutrìand and N. Tchou,
Hamilton-Jacobi equations constrained on networks, NoDEA Nonlinear Differential Equations Appl., 20 (2013), 413-445.
doi: 10.1007/s00030-012-0158-1. |
[3] |
M. G. Crandall, H. Ishii and P.-L. Lions,
User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.), 27 (1992), 1-67.
doi: 10.1090/S0273-0979-1992-00266-5. |
[4] |
M. G. Crandall and P.-L. Lions,
Viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc., 277 (1983), 1-42.
doi: 10.1090/S0002-9947-1983-0690039-8. |
[5] |
L. C. Evans,
The perturbed test function method for viscosity solutions of nonlinear PDE, Proc. Roy. Soc. Edinburgh Sect. A, 111 (1989), 359-375.
doi: 10.1017/S0308210500018631. |
[6] |
M. I. Freidlin and A. D. Wentzell, Random perturbations of Hamiltonian systems, Mem. Amer. Math. Soc., 109 (1994), viii+82pp.
doi: 10.1090/memo/0523. |
[7] |
G. Galise, C. Imbert and R. Monneau,
A junction condition by specified homogenization and application to traffic lights, Anal. PDE, 8 (2015), 1891-1929.
doi: 10.2140/apde.2015.8.1891. |
[8] |
C. Imbert and R. Monneau, Flux-limited solutions for quasi-convex Hamilton-Jacobi equations on networks, Ann. Sci. Éc. Norm. Supér. (4), 50 (2017), 357–448.
doi: 10.24033/asens.2323. |
[9] |
C. Imbert, R. Monneau and H. Zidani,
A Hamilton-Jacobi approach to junction problems and application to traffic flows, ESAIM Control Optim. Calc. Var., 19 (2013), 129-166.
doi: 10.1051/cocv/2012002. |
[10] |
H. Ishii,
Perron's method for Hamilton-Jacobi equations, Duke Math. J., 55 (1987), 369-384.
doi: 10.1215/S0012-7094-87-05521-9. |
[11] |
H. Ishii,
A boundary value problem of the Dirichlet type for Hamilton-Jacobi equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 16 (1989), 105-135.
|
[12] |
H. Ishii and P. E. Souganidis,
A pde approach to small stochastic perturbations of Hamiltonian flows, J. Differential Equations, 252 (2012), 1748-1775.
doi: 10.1016/j.jde.2011.08.036. |
[13] |
T. Kumagai,
A perturbation problem involving singular perturbations of domains for Hamilton-Jacobi equations, Funkcial. Ekvac., 61 (2018), 377-427.
doi: 10.1619/fesi.61.377. |
[14] |
T. Kumagai, An asymptotic analysis for Hamilton-Jacobi equations with large Hamiltonian drift terms, Adv. Calc. Var., 2017.
doi: 10.1515/acv-2017-0046. |
[15] |
T. Kumagai, A Study of Hamilton-Jacobi Equations with Large Hamiltonian Drift Terms, Ph.D, Waseda University, Tokyo, Japan, 2018. |
[16] |
P.-L. Lions and P. Souganidis,
Viscosity solutions for junctions: Well posedness and stability, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl., 27 (2016), 535-545.
doi: 10.4171/RLM/747. |
[17] |
P.-L. Lions and P. Souganidis,
Well-posedness for multi-dimensional junction problems with Kirchoff-type conditions, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl., 28 (2017), 807-816.
doi: 10.4171/RLM/786. |
show all references
References:
[1] |
Y. Achdou and N. Tchou,
Hamilton-Jacobi equations on networks as limits of singularly perturbed problems in optimal control: dimension reduction, Comm. Partial Differential Equations, 40 (2015), 652-693.
doi: 10.1080/03605302.2014.974764. |
[2] |
Y. Achdou, F. Camilli, A. Cutrìand and N. Tchou,
Hamilton-Jacobi equations constrained on networks, NoDEA Nonlinear Differential Equations Appl., 20 (2013), 413-445.
doi: 10.1007/s00030-012-0158-1. |
[3] |
M. G. Crandall, H. Ishii and P.-L. Lions,
User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.), 27 (1992), 1-67.
doi: 10.1090/S0273-0979-1992-00266-5. |
[4] |
M. G. Crandall and P.-L. Lions,
Viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc., 277 (1983), 1-42.
doi: 10.1090/S0002-9947-1983-0690039-8. |
[5] |
L. C. Evans,
The perturbed test function method for viscosity solutions of nonlinear PDE, Proc. Roy. Soc. Edinburgh Sect. A, 111 (1989), 359-375.
doi: 10.1017/S0308210500018631. |
[6] |
M. I. Freidlin and A. D. Wentzell, Random perturbations of Hamiltonian systems, Mem. Amer. Math. Soc., 109 (1994), viii+82pp.
doi: 10.1090/memo/0523. |
[7] |
G. Galise, C. Imbert and R. Monneau,
A junction condition by specified homogenization and application to traffic lights, Anal. PDE, 8 (2015), 1891-1929.
doi: 10.2140/apde.2015.8.1891. |
[8] |
C. Imbert and R. Monneau, Flux-limited solutions for quasi-convex Hamilton-Jacobi equations on networks, Ann. Sci. Éc. Norm. Supér. (4), 50 (2017), 357–448.
doi: 10.24033/asens.2323. |
[9] |
C. Imbert, R. Monneau and H. Zidani,
A Hamilton-Jacobi approach to junction problems and application to traffic flows, ESAIM Control Optim. Calc. Var., 19 (2013), 129-166.
doi: 10.1051/cocv/2012002. |
[10] |
H. Ishii,
Perron's method for Hamilton-Jacobi equations, Duke Math. J., 55 (1987), 369-384.
doi: 10.1215/S0012-7094-87-05521-9. |
[11] |
H. Ishii,
A boundary value problem of the Dirichlet type for Hamilton-Jacobi equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 16 (1989), 105-135.
|
[12] |
H. Ishii and P. E. Souganidis,
A pde approach to small stochastic perturbations of Hamiltonian flows, J. Differential Equations, 252 (2012), 1748-1775.
doi: 10.1016/j.jde.2011.08.036. |
[13] |
T. Kumagai,
A perturbation problem involving singular perturbations of domains for Hamilton-Jacobi equations, Funkcial. Ekvac., 61 (2018), 377-427.
doi: 10.1619/fesi.61.377. |
[14] |
T. Kumagai, An asymptotic analysis for Hamilton-Jacobi equations with large Hamiltonian drift terms, Adv. Calc. Var., 2017.
doi: 10.1515/acv-2017-0046. |
[15] |
T. Kumagai, A Study of Hamilton-Jacobi Equations with Large Hamiltonian Drift Terms, Ph.D, Waseda University, Tokyo, Japan, 2018. |
[16] |
P.-L. Lions and P. Souganidis,
Viscosity solutions for junctions: Well posedness and stability, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl., 27 (2016), 535-545.
doi: 10.4171/RLM/747. |
[17] |
P.-L. Lions and P. Souganidis,
Well-posedness for multi-dimensional junction problems with Kirchoff-type conditions, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl., 28 (2017), 807-816.
doi: 10.4171/RLM/786. |

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