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A comparison principle for Hamilton-Jacobi equation with moving in time boundary
1. | Normandie Univ, INSA de Rouen, LMI (EA 3226 - FR CNRS 3335), 76000 Rouen, France |
2. | 685 Avenue de l'Université, 76801 St Etienne du Rouvray cedex, France |
In this paper we consider an Hamilton-Jacobi equation on a moving in time domain. The boundary is described by a $C^{1}$ function. We show how we derive this equation from the work of [
References:
[1] |
Y. Achdou, C. Camilli, A. Cutrì and N. Tchou,
Hamilton–Jacobi equations constrained on networks, Nonlinear Differential Equations and Applications NoDEA, 20 (2013), 413-445.
doi: 10.1007/s00030-012-0158-1. |
[2] |
G. Barles,
Nonlinear neumann boundary conditions for quasilinear degenerate elliptic equations and applications, Journal of Differential Equations, 154 (1999), 191-224.
doi: 10.1006/jdeq.1998.3568. |
[3] |
G. Barles, An introduction to the theory of viscosity solutions for first-order hamilton-jacobi equations and applications, in Hamilton-Jacobi Equations: Approximations, Numerical Analysis and Applications, Springer, 2074 (2013), 49–109.
doi: 10.1007/978-3-642-36433-4_2. |
[4] |
G. Barles, A. Briani, E. Chasseigne and C. Imbert, Flux-limited and classical viscosity solutions for regional control problems, ESAIM Control Optim. Calc. Var., 24 (2018), 1881–1906, arXiv: 1611.01977.
doi: 10.1051/cocv/2017076. |
[5] |
G. Barles and P. Lions,
Fully nonlinear Neumann type boundary conditions for first-order Hamilton–Jacobi equations, Nonlinear Analysis: Theory, Methods & Applications, 16 (1991), 143-153.
doi: 10.1016/0362-546X(91)90165-W. |
[6] |
R. Borsche, R. Colombo and M. Garavello,
On the coupling of systems of hyperbolic conservation laws with ordinary differential equations, Nonlinearity, 23 (2010), 2749-2770.
doi: 10.1088/0951-7715/23/11/002. |
[7] |
R. Borsche, R. Colombo and M. Garavello,
Mixed systems: ODEs–balance laws, Journal of Differential equations, 252 (2012), 2311-2338.
doi: 10.1016/j.jde.2011.08.051. |
[8] |
G. Coclite and M. Garavello,
Vanishing viscosity for mixed systems with moving boundaries, Journal of Functional Analysis, 264 (2013), 1664-1710.
doi: 10.1016/j.jfa.2013.01.010. |
[9] |
R. M. Colombo and P. Goatin,
A well posed conservation law with a variable unilateral constraint, Journal of Differential Equations, 234 (2007), 654-675.
doi: 10.1016/j.jde.2006.10.014. |
[10] |
M. G. Crandall, H. Ishii and P.-L. Lions,
User's guide to viscosity solutions of second order partial differential equations, Bulletin of the American Mathematical Society, 27 (1992), 1-67.
doi: 10.1090/S0273-0979-1992-00266-5. |
[11] |
C. F. Daganzo and J. A. Laval,
Moving bottlenecks: A numerical method that converges in flows, Transportation Research Part B: Methodological, 39 (2005), 855-863.
doi: 10.1016/j.trb.2004.10.004. |
[12] |
M. L. Delle Monache and P. Goatin,
Scalar conservation laws with moving constraints arising in traffic flow modeling: An existence result, Journal of Differential Equations, 257 (2015), 4015-4029.
doi: 10.1016/j.jde.2014.07.014. |
[13] |
A. Fino, H. Ibrahim and R. Monneau,
The Peierls–Nabarro model as a limit of a Frenkel–Kontorova model, Journal of Differential Equations, 252 (2012), 258-293.
doi: 10.1016/j.jde.2011.08.007. |
[14] |
N. Forcadel, W. Salazar and M. Zaydan,
Homogenization of second order discrete model with local perturbation and application to traffic flow, Discrete and Continuous Dynamical Systems-Series A, 37 (2017), 1437-1487.
doi: 10.3934/dcds.2017060. |
[15] |
N. Forcadel, W. Salazar and M. Zaydan,
Specified homogenization of a discrete traffic model leading to an effective junction condition, Communications on Pure & Applied Analysis, 17 (2018), 2173-2206.
doi: 10.3934/cpaa.2018104. |
[16] |
G. Galise, C. Imbert and R. Monneau,
A junction condition by specified homogenization and application to traffic lights, Analysis & PDE, 8 (2015), 1891-1929.
doi: 10.2140/apde.2015.8.1891. |
[17] |
M. Garavello, R. Natalini, B. Piccoli and A. Terracina,
Conservation laws with discontinuous flux, NHM, 2 (2007), 159-179.
doi: 10.3934/nhm.2007.2.159. |
[18] |
B. Greenshields, Ws. Channing, H. Miller and others, A Study of Traffic Capacity, , Highway research board proceedings, 1935. |
[19] |
J. Guerand, Classification of nonlinear boundary conditions for 1D nonconvex Hamilton-Jacobi equations, arXiv: 1609.08867, 2016. |
[20] |
C. Imbert and R. Monneau,
Flux-limited solutions for quasi-convex Hamilton-Jacobi equations on networks, Annales Scientifiques de l'ENS, 50 (2017), 357-448.
doi: 10.24033/asens.2323. |
[21] |
C. Imbert and R. Monneau,
Quasi-convex Hamilton-Jacobi equations posed on junctions: The multi-dimensional case, Discrete and Continuous Dynamical Systems-Series A, 37 (2017), 6405-6435.
doi: 10.3934/dcds.2017278. |
[22] |
C. Imbert, R. Monneau and H. Zidani,
A Hamilton-Jacobi approach to junction problems and application to traffic flows, ESAIM: Control, Optimisation and Calculus of Variations, 19 (2013), 129-166.
doi: 10.1051/cocv/2012002. |
[23] |
H. Ishii and ot hers,
Fully nonlinear oblique derivative problems for nonlinear second-order elliptic PDEs, Duke Math. J, 62 (1991), 633-661.
doi: 10.1215/S0012-7094-91-06228-9. |
[24] |
C. Lattanzio, A. Maurizi and B. Piccoli,
Moving bottlenecks in car traffic flow: A PDE-ODE coupled model, SIAM Journal on Mathematical Analysis, 43 (2011), 50-67.
doi: 10.1137/090767224. |
[25] |
J. Lebacque and M. Khoshyaran, Modelling vehicular traffic flow on networks using macroscopic models, in Finite Volumes for Complex Applications II, (1999), 551–558. |
[26] |
J. Lebacque, and others, Introducing buses into first-order macroscopic traffic flow models, Transportation Research Record: Journal of the Transportation Research Board, (1998), 70–79.
doi: 10.3141/1644-08. |
[27] |
P.-L. Lions and P. Souganidis,
Viscosity solutions for junctions: well posedness and stability, Rendiconti Lincei-Matematica E Applicazioni, 27 (2016), 535-545.
doi: 10.4171/RLM/747. |
[28] |
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, arXiv: 1704.04001.
doi: 10.4171/RLM/786. |
show all references
References:
[1] |
Y. Achdou, C. Camilli, A. Cutrì and N. Tchou,
Hamilton–Jacobi equations constrained on networks, Nonlinear Differential Equations and Applications NoDEA, 20 (2013), 413-445.
doi: 10.1007/s00030-012-0158-1. |
[2] |
G. Barles,
Nonlinear neumann boundary conditions for quasilinear degenerate elliptic equations and applications, Journal of Differential Equations, 154 (1999), 191-224.
doi: 10.1006/jdeq.1998.3568. |
[3] |
G. Barles, An introduction to the theory of viscosity solutions for first-order hamilton-jacobi equations and applications, in Hamilton-Jacobi Equations: Approximations, Numerical Analysis and Applications, Springer, 2074 (2013), 49–109.
doi: 10.1007/978-3-642-36433-4_2. |
[4] |
G. Barles, A. Briani, E. Chasseigne and C. Imbert, Flux-limited and classical viscosity solutions for regional control problems, ESAIM Control Optim. Calc. Var., 24 (2018), 1881–1906, arXiv: 1611.01977.
doi: 10.1051/cocv/2017076. |
[5] |
G. Barles and P. Lions,
Fully nonlinear Neumann type boundary conditions for first-order Hamilton–Jacobi equations, Nonlinear Analysis: Theory, Methods & Applications, 16 (1991), 143-153.
doi: 10.1016/0362-546X(91)90165-W. |
[6] |
R. Borsche, R. Colombo and M. Garavello,
On the coupling of systems of hyperbolic conservation laws with ordinary differential equations, Nonlinearity, 23 (2010), 2749-2770.
doi: 10.1088/0951-7715/23/11/002. |
[7] |
R. Borsche, R. Colombo and M. Garavello,
Mixed systems: ODEs–balance laws, Journal of Differential equations, 252 (2012), 2311-2338.
doi: 10.1016/j.jde.2011.08.051. |
[8] |
G. Coclite and M. Garavello,
Vanishing viscosity for mixed systems with moving boundaries, Journal of Functional Analysis, 264 (2013), 1664-1710.
doi: 10.1016/j.jfa.2013.01.010. |
[9] |
R. M. Colombo and P. Goatin,
A well posed conservation law with a variable unilateral constraint, Journal of Differential Equations, 234 (2007), 654-675.
doi: 10.1016/j.jde.2006.10.014. |
[10] |
M. G. Crandall, H. Ishii and P.-L. Lions,
User's guide to viscosity solutions of second order partial differential equations, Bulletin of the American Mathematical Society, 27 (1992), 1-67.
doi: 10.1090/S0273-0979-1992-00266-5. |
[11] |
C. F. Daganzo and J. A. Laval,
Moving bottlenecks: A numerical method that converges in flows, Transportation Research Part B: Methodological, 39 (2005), 855-863.
doi: 10.1016/j.trb.2004.10.004. |
[12] |
M. L. Delle Monache and P. Goatin,
Scalar conservation laws with moving constraints arising in traffic flow modeling: An existence result, Journal of Differential Equations, 257 (2015), 4015-4029.
doi: 10.1016/j.jde.2014.07.014. |
[13] |
A. Fino, H. Ibrahim and R. Monneau,
The Peierls–Nabarro model as a limit of a Frenkel–Kontorova model, Journal of Differential Equations, 252 (2012), 258-293.
doi: 10.1016/j.jde.2011.08.007. |
[14] |
N. Forcadel, W. Salazar and M. Zaydan,
Homogenization of second order discrete model with local perturbation and application to traffic flow, Discrete and Continuous Dynamical Systems-Series A, 37 (2017), 1437-1487.
doi: 10.3934/dcds.2017060. |
[15] |
N. Forcadel, W. Salazar and M. Zaydan,
Specified homogenization of a discrete traffic model leading to an effective junction condition, Communications on Pure & Applied Analysis, 17 (2018), 2173-2206.
doi: 10.3934/cpaa.2018104. |
[16] |
G. Galise, C. Imbert and R. Monneau,
A junction condition by specified homogenization and application to traffic lights, Analysis & PDE, 8 (2015), 1891-1929.
doi: 10.2140/apde.2015.8.1891. |
[17] |
M. Garavello, R. Natalini, B. Piccoli and A. Terracina,
Conservation laws with discontinuous flux, NHM, 2 (2007), 159-179.
doi: 10.3934/nhm.2007.2.159. |
[18] |
B. Greenshields, Ws. Channing, H. Miller and others, A Study of Traffic Capacity, , Highway research board proceedings, 1935. |
[19] |
J. Guerand, Classification of nonlinear boundary conditions for 1D nonconvex Hamilton-Jacobi equations, arXiv: 1609.08867, 2016. |
[20] |
C. Imbert and R. Monneau,
Flux-limited solutions for quasi-convex Hamilton-Jacobi equations on networks, Annales Scientifiques de l'ENS, 50 (2017), 357-448.
doi: 10.24033/asens.2323. |
[21] |
C. Imbert and R. Monneau,
Quasi-convex Hamilton-Jacobi equations posed on junctions: The multi-dimensional case, Discrete and Continuous Dynamical Systems-Series A, 37 (2017), 6405-6435.
doi: 10.3934/dcds.2017278. |
[22] |
C. Imbert, R. Monneau and H. Zidani,
A Hamilton-Jacobi approach to junction problems and application to traffic flows, ESAIM: Control, Optimisation and Calculus of Variations, 19 (2013), 129-166.
doi: 10.1051/cocv/2012002. |
[23] |
H. Ishii and ot hers,
Fully nonlinear oblique derivative problems for nonlinear second-order elliptic PDEs, Duke Math. J, 62 (1991), 633-661.
doi: 10.1215/S0012-7094-91-06228-9. |
[24] |
C. Lattanzio, A. Maurizi and B. Piccoli,
Moving bottlenecks in car traffic flow: A PDE-ODE coupled model, SIAM Journal on Mathematical Analysis, 43 (2011), 50-67.
doi: 10.1137/090767224. |
[25] |
J. Lebacque and M. Khoshyaran, Modelling vehicular traffic flow on networks using macroscopic models, in Finite Volumes for Complex Applications II, (1999), 551–558. |
[26] |
J. Lebacque, and others, Introducing buses into first-order macroscopic traffic flow models, Transportation Research Record: Journal of the Transportation Research Board, (1998), 70–79.
doi: 10.3141/1644-08. |
[27] |
P.-L. Lions and P. Souganidis,
Viscosity solutions for junctions: well posedness and stability, Rendiconti Lincei-Matematica E Applicazioni, 27 (2016), 535-545.
doi: 10.4171/RLM/747. |
[28] |
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, arXiv: 1704.04001.
doi: 10.4171/RLM/786. |


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