American Institute of Mathematical Sciences

Advanced
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 and 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.

[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.

Figure 1.  $ N = 6 $
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Claudio Marchi. On the convergence of singular perturbations of Hamilton-Jacobi equations. Communications on Pure and Applied Analysis, 2010, 9 (5) : 1363-1377. doi: 10.3934/cpaa.2010.9.1363
[2]

Sergey Rashkovskiy. Hamilton-Jacobi theory for Hamiltonian and non-Hamiltonian systems. Journal of Geometric Mechanics, 2020, 12 (4) : 563-583. doi: 10.3934/jgm.2020024
[3]

Isabeau Birindelli, J. Wigniolle. Homogenization of Hamilton-Jacobi equations in the Heisenberg group. Communications on Pure and Applied Analysis, 2003, 2 (4) : 461-479. doi: 10.3934/cpaa.2003.2.461
[4]

Gonzalo Dávila. Comparison principles for nonlocal Hamilton-Jacobi equations. Discrete and Continuous Dynamical Systems, 2022  doi: 10.3934/dcds.2022061
[5]

Laura Caravenna, Annalisa Cesaroni, Hung Vinh Tran. Preface: Recent developments related to conservation laws and Hamilton-Jacobi equations. Discrete and Continuous Dynamical Systems - S, 2018, 11 (5) : i-iii. doi: 10.3934/dcdss.201805i
[6]

Fabio Camilli, Paola Loreti, Naoki Yamada. Systems of convex Hamilton-Jacobi equations with implicit obstacles and the obstacle problem. Communications on Pure and Applied Analysis, 2009, 8 (4) : 1291-1302. doi: 10.3934/cpaa.2009.8.1291
[7]

Yasuhiro Fujita, Katsushi Ohmori. Inequalities and the Aubry-Mather theory of Hamilton-Jacobi equations. Communications on Pure and Applied Analysis, 2009, 8 (2) : 683-688. doi: 10.3934/cpaa.2009.8.683
[8]

Olga Bernardi, Franco Cardin. On $C^0$-variational solutions for Hamilton-Jacobi equations. Discrete and Continuous Dynamical Systems, 2011, 31 (2) : 385-406. doi: 10.3934/dcds.2011.31.385
[9]

Emeric Bouin. A Hamilton-Jacobi approach for front propagation in kinetic equations. Kinetic and Related Models, 2015, 8 (2) : 255-280. doi: 10.3934/krm.2015.8.255
[10]

Gawtum Namah, Mohammed Sbihi. A notion of extremal solutions for time periodic Hamilton-Jacobi equations. Discrete and Continuous Dynamical Systems - B, 2010, 13 (3) : 647-664. doi: 10.3934/dcdsb.2010.13.647
[11]

Antonio Avantaggiati, Paola Loreti, Cristina Pocci. Mixed norms, functional Inequalities, and Hamilton-Jacobi equations. Discrete and Continuous Dynamical Systems - B, 2014, 19 (7) : 1855-1867. doi: 10.3934/dcdsb.2014.19.1855
[12]

Martino Bardi, Yoshikazu Giga. Right accessibility of semicontinuous initial data for Hamilton-Jacobi equations. Communications on Pure and Applied Analysis, 2003, 2 (4) : 447-459. doi: 10.3934/cpaa.2003.2.447
[13]

Xifeng Su, Lin Wang, Jun Yan. Weak KAM theory for HAMILTON-JACOBI equations depending on unknown functions. Discrete and Continuous Dynamical Systems, 2016, 36 (11) : 6487-6522. doi: 10.3934/dcds.2016080
[14]

Gui-Qiang Chen, Bo Su. Discontinuous solutions for Hamilton-Jacobi equations: Uniqueness and regularity. Discrete and Continuous Dynamical Systems, 2003, 9 (1) : 167-192. doi: 10.3934/dcds.2003.9.167
[15]

David McCaffrey. A representational formula for variational solutions to Hamilton-Jacobi equations. Communications on Pure and Applied Analysis, 2012, 11 (3) : 1205-1215. doi: 10.3934/cpaa.2012.11.1205
[16]

Mihai Bostan, Gawtum Namah. Time periodic viscosity solutions of Hamilton-Jacobi equations. Communications on Pure and Applied Analysis, 2007, 6 (2) : 389-410. doi: 10.3934/cpaa.2007.6.389
[17]

Piermarco Cannarsa, Marco Mazzola, Carlo Sinestrari. Global propagation of singularities for time dependent Hamilton-Jacobi equations. Discrete and Continuous Dynamical Systems, 2015, 35 (9) : 4225-4239. doi: 10.3934/dcds.2015.35.4225
[18]

Qing Liu, Atsushi Nakayasu. Convexity preserving properties for Hamilton-Jacobi equations in geodesic spaces. Discrete and Continuous Dynamical Systems, 2019, 39 (1) : 157-183. doi: 10.3934/dcds.2019007
[19]

Olga Bernardi, Franco Cardin. Minimax and viscosity solutions of Hamilton-Jacobi equations in the convex case. Communications on Pure and Applied Analysis, 2006, 5 (4) : 793-812. doi: 10.3934/cpaa.2006.5.793
[20]

Kaizhi Wang, Jun Yan. Lipschitz dependence of viscosity solutions of Hamilton-Jacobi equations with respect to the parameter. Discrete and Continuous Dynamical Systems, 2016, 36 (3) : 1649-1659. doi: 10.3934/dcds.2016.36.1649

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (209)
  • HTML views (199)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy