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April  2021, 41(4): 1519-1542. doi: 10.3934/dcds.2020329

Averaging of Hamilton-Jacobi equations along divergence-free vector fields

Hitoshi Ishii 1,, and  Taiga Kumagai 2,

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

* Corresponding author: Hitoshi Ishii

Received  January 2020 Revised  June 2020 Published  April 2021 Early access  September 2020

Fund Project: The first author is partially supported by the JSPS grants: KAKENHI #16H03948, #18H00833, #20K03688, #20H01817. The second author is partially supported by the JSPS grants: KAKENHI #20K03688

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 $.

Keywords: Hamilton-Jacobi equations, averaging, Hamiltonian drifts, singular perturbations.
Mathematics Subject Classification: Primary: 35B25, 35F21; Secondary: 35F30, 35D40, 49L25.
Citation: Hitoshi Ishii, Taiga Kumagai. Averaging of Hamilton-Jacobi equations along divergence-free vector fields. Discrete & Continuous Dynamical Systems, 2021, 41 (4) : 1519-1542. doi: 10.3934/dcds.2020329
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

T. Kumagai, A Study of Hamilton-Jacobi Equations with Large Hamiltonian Drift Terms, Ph.D, Waseda University, Tokyo, Japan, 2018. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

T. Kumagai, A Study of Hamilton-Jacobi Equations with Large Hamiltonian Drift Terms, Ph.D, Waseda University, Tokyo, Japan, 2018. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

Figure 1.  $ N = 6 $
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