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September  2019, 8(3): 543-565. doi: 10.3934/eect.2019026

A comparison principle for Hamilton-Jacobi equation with moving in time boundary

Nicolas Forcadel 1,2,, and  Mamdouh Zaydan 1,2,

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

 

Received  May 2018 Revised  November 2018 Published  September 2019 Early access  May 2019

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 [26]. We only prove a comparison principle since the proof of other theoritical results can be found in [20]. At the end of the paper, we consider a short homogenization result in order to reinforce the traffic flow interpretation of the equation.

Keywords: Comparison principle, viscosity solutions, Hamilton-Jacobi equations, junction conditions, macroscopic traffic model.
Mathematics Subject Classification: 35D40, 90B20, 35B27, 35F20, 45K05.
Citation: Nicolas Forcadel, Mamdouh Zaydan. A comparison principle for Hamilton-Jacobi equation with moving in time boundary. Evolution Equations & Control Theory, 2019, 8 (3) : 543-565. doi: 10.3934/eect.2019026
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

B. Greenshields, Ws. Channing, H. Miller and others, A Study of Traffic Capacity, , Highway research board proceedings, 1935. Google Scholar

[19]

J. Guerand, Classification of nonlinear boundary conditions for 1D nonconvex Hamilton-Jacobi equations, arXiv: 1609.08867, 2016. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

B. Greenshields, Ws. Channing, H. Miller and others, A Study of Traffic Capacity, , Highway research board proceedings, 1935. Google Scholar

[19]

J. Guerand, Classification of nonlinear boundary conditions for 1D nonconvex Hamilton-Jacobi equations, arXiv: 1609.08867, 2016. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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