
-
Previous Article
Measure-theoretic Lie brackets for nonsmooth vector fields
- DCDS-S Home
- This Issue
-
Next Article
The vanishing viscosity limit for a system of H-J equations related to a debt management problem
A flame propagation model on a network with application to a blocking problem
1. | Dip. di Scienze di Base e Applicate per l'Ingegneria, "Sapienza" Università di Roma, via Scarpa 16, 00161 Roma, Italy |
2. | Dipartimento di Matematica, "Sapienza" Università di Roma, p.le A. Moro 5, 00185 Roma, Italy |
3. | Dip. di Ingegneria dell'Informazione, Università di Padova, via Gradenigo 6/B, 35131 Padova, Italy |
$\left\{ \begin{array}{*{35}{l}} {{\partial }_{t}}u+H(x,Du) = 0&(x,t)\in \Gamma \times (0,T) \\ u(x,0) = {{u}_{0}}(x)&x\in \Gamma \\\end{array} \right.$ |
$\Gamma$ |
$H$ |
References:
[1] |
Y. Achdou, F. Camilli, A. Cutrí 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. |
[2] |
M. Bardi and I. Capuzzo Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations, Birkäuser, Boston, 1997.
doi: 10.1007/978-0-8176-4755-1. |
[3] |
G. Barles, Remark on a Flame Propagation Model, Report INRIA #464, 1985. |
[4] |
G. Barles, H. M. Soner and P. Souganidis,
Front propagations and phase field theory, SIAM J. Control Optim., 31 (1993), 439-469.
doi: 10.1137/0331021. |
[5] |
F. Camilli and C. Marchi,
A comparison among various notions of viscosity solution for Hamilton-Jacobi equations on networks, J. Math. Anal. Appl., 407 (2013), 112-118.
doi: 10.1016/j.jmaa.2013.05.015. |
[6] |
F. Camilli, C. Marchi and D. Schieborn,
The vanishing viscosity limit for Hamilton-Jacobi equation on networks, J. Differential Equations, 254 (2013), 4122-4143.
doi: 10.1016/j.jde.2013.02.013. |
[7] |
F. Camilli, A. Festa and D. Schieborn,
An approximation scheme for an Hamilton-Jacobi equation defined on a network, Applied Num. Math., 73 (2013), 33-47.
doi: 10.1016/j.apnum.2013.05.003. |
[8] |
Y. G. Chen, Y. Giga and S. Goto,
Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations, J. Differential Geom., 33 (1991), 749-786.
doi: 10.4310/jdg/1214446564. |
[9] |
G. Costeseque, J.-P. Lebacque and R. Monneau,
A convergent scheme for Hamilton-Jacobi equations on a junction: application to traffic, Numer. Math., 129 (2015), 405-447.
doi: 10.1007/s00211-014-0643-z. |
[10] |
M. Garavello and B. Piccoli, Traffic Flow on Networks, AIMS Series on Applied Mathematics, American Institute of Mathematical Sciences, Springfield, MO, 2006. |
[11] |
A. Khanafer and T. Başar, Information Spread in Networks: Control, Game and Equilibria, Proc. Information theory and Application Workshop (ITA'14), San Diego, 2014. |
[12] |
C. Imbert and R. Monneau,
Flux-limited solutions for quasi-convex Hamilton-Jacobi equations on networks, Ann. Sci. Éc. Norm. Supér., 50 (2017), 357-448.
doi: 10.24033/asens.2323. |
[13] |
P.-L. Lions and P. E. 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. |
[14] |
P.-L. Lions and P. E. 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. |
[15] |
D. Mugnolo, Semigroup Methods for Evolution Equations on Networks, Understanding Complex Systems, Springer, Berlin, 2014.
doi: 10.1007/978-3-319-04621-1. |
[16] |
G. Namah and J. M. Roquejoffre,
Remarks on the long time behaviour of the solutions of Hamilton-Jacobi equations, Comm. Partial Differential Equations, 24 (1999), 883-893.
doi: 10.1080/03605309908821451. |
[17] |
D. Schieborn and F. Camilli,
Viscosity solutions of Eikonal equations on topological network, Calc. Var. Partial Differential Equations, 46 (2013), 671-686.
doi: 10.1007/s00526-012-0498-z. |
[18] |
A. Siconolfi,
A first order Hamilton-Jacobi equation with singularity and the evolution of level sets, Comm. Partial Differential Equations, 20 (1995), 277-307.
doi: 10.1080/03605309508821094. |
[19] |
A. Siconolfi,
Metric character of Hamilton-Jacobi equations, Trans. Amer. Math. Soc., 355 (2003), 1987-2009.
doi: 10.1090/S0002-9947-03-03237-9. |
[20] |
P. Soravia,
Generalized motion of front along its normal direction: A differential game approach, Nonlinear Anal. TMA, 22 (1994), 1247-1262.
doi: 10.1016/0362-546X(94)90108-2. |
[21] |
P. Van Mieghem, J. Omic and R. Kooij,
Virus spread in Networks, IEEE/ACM Trans. on networking, 17 (2009), 1-14.
doi: 10.1109/TNET.2008.925623. |
show all references
References:
[1] |
Y. Achdou, F. Camilli, A. Cutrí 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. |
[2] |
M. Bardi and I. Capuzzo Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations, Birkäuser, Boston, 1997.
doi: 10.1007/978-0-8176-4755-1. |
[3] |
G. Barles, Remark on a Flame Propagation Model, Report INRIA #464, 1985. |
[4] |
G. Barles, H. M. Soner and P. Souganidis,
Front propagations and phase field theory, SIAM J. Control Optim., 31 (1993), 439-469.
doi: 10.1137/0331021. |
[5] |
F. Camilli and C. Marchi,
A comparison among various notions of viscosity solution for Hamilton-Jacobi equations on networks, J. Math. Anal. Appl., 407 (2013), 112-118.
doi: 10.1016/j.jmaa.2013.05.015. |
[6] |
F. Camilli, C. Marchi and D. Schieborn,
The vanishing viscosity limit for Hamilton-Jacobi equation on networks, J. Differential Equations, 254 (2013), 4122-4143.
doi: 10.1016/j.jde.2013.02.013. |
[7] |
F. Camilli, A. Festa and D. Schieborn,
An approximation scheme for an Hamilton-Jacobi equation defined on a network, Applied Num. Math., 73 (2013), 33-47.
doi: 10.1016/j.apnum.2013.05.003. |
[8] |
Y. G. Chen, Y. Giga and S. Goto,
Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations, J. Differential Geom., 33 (1991), 749-786.
doi: 10.4310/jdg/1214446564. |
[9] |
G. Costeseque, J.-P. Lebacque and R. Monneau,
A convergent scheme for Hamilton-Jacobi equations on a junction: application to traffic, Numer. Math., 129 (2015), 405-447.
doi: 10.1007/s00211-014-0643-z. |
[10] |
M. Garavello and B. Piccoli, Traffic Flow on Networks, AIMS Series on Applied Mathematics, American Institute of Mathematical Sciences, Springfield, MO, 2006. |
[11] |
A. Khanafer and T. Başar, Information Spread in Networks: Control, Game and Equilibria, Proc. Information theory and Application Workshop (ITA'14), San Diego, 2014. |
[12] |
C. Imbert and R. Monneau,
Flux-limited solutions for quasi-convex Hamilton-Jacobi equations on networks, Ann. Sci. Éc. Norm. Supér., 50 (2017), 357-448.
doi: 10.24033/asens.2323. |
[13] |
P.-L. Lions and P. E. 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. |
[14] |
P.-L. Lions and P. E. 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. |
[15] |
D. Mugnolo, Semigroup Methods for Evolution Equations on Networks, Understanding Complex Systems, Springer, Berlin, 2014.
doi: 10.1007/978-3-319-04621-1. |
[16] |
G. Namah and J. M. Roquejoffre,
Remarks on the long time behaviour of the solutions of Hamilton-Jacobi equations, Comm. Partial Differential Equations, 24 (1999), 883-893.
doi: 10.1080/03605309908821451. |
[17] |
D. Schieborn and F. Camilli,
Viscosity solutions of Eikonal equations on topological network, Calc. Var. Partial Differential Equations, 46 (2013), 671-686.
doi: 10.1007/s00526-012-0498-z. |
[18] |
A. Siconolfi,
A first order Hamilton-Jacobi equation with singularity and the evolution of level sets, Comm. Partial Differential Equations, 20 (1995), 277-307.
doi: 10.1080/03605309508821094. |
[19] |
A. Siconolfi,
Metric character of Hamilton-Jacobi equations, Trans. Amer. Math. Soc., 355 (2003), 1987-2009.
doi: 10.1090/S0002-9947-03-03237-9. |
[20] |
P. Soravia,
Generalized motion of front along its normal direction: A differential game approach, Nonlinear Anal. TMA, 22 (1994), 1247-1262.
doi: 10.1016/0362-546X(94)90108-2. |
[21] |
P. Van Mieghem, J. Omic and R. Kooij,
Virus spread in Networks, IEEE/ACM Trans. on networking, 17 (2009), 1-14.
doi: 10.1109/TNET.2008.925623. |









[1] |
Federica Dragoni. Metric Hopf-Lax formula with semicontinuous data. Discrete and Continuous Dynamical Systems, 2007, 17 (4) : 713-729. doi: 10.3934/dcds.2007.17.713 |
[2] |
Juan Pablo Rincón-Zapatero. Hopf-Lax formula for variational problems with non-constant discount. Journal of Geometric Mechanics, 2009, 1 (3) : 357-367. doi: 10.3934/jgm.2009.1.357 |
[3] |
Joan-Andreu Lázaro-Camí, Juan-Pablo Ortega. The stochastic Hamilton-Jacobi equation. Journal of Geometric Mechanics, 2009, 1 (3) : 295-315. doi: 10.3934/jgm.2009.1.295 |
[4] |
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 |
[5] |
Tomoki Ohsawa, Anthony M. Bloch. Nonholonomic Hamilton-Jacobi equation and integrability. Journal of Geometric Mechanics, 2009, 1 (4) : 461-481. doi: 10.3934/jgm.2009.1.461 |
[6] |
Nalini Anantharaman, Renato Iturriaga, Pablo Padilla, Héctor Sánchez-Morgado. Physical solutions of the Hamilton-Jacobi equation. Discrete and Continuous Dynamical Systems - B, 2005, 5 (3) : 513-528. doi: 10.3934/dcdsb.2005.5.513 |
[7] |
María Barbero-Liñán, Manuel de León, David Martín de Diego, Juan C. Marrero, Miguel C. Muñoz-Lecanda. Kinematic reduction and the Hamilton-Jacobi equation. Journal of Geometric Mechanics, 2012, 4 (3) : 207-237. doi: 10.3934/jgm.2012.4.207 |
[8] |
Larry M. Bates, Francesco Fassò, Nicola Sansonetto. The Hamilton-Jacobi equation, integrability, and nonholonomic systems. Journal of Geometric Mechanics, 2014, 6 (4) : 441-449. doi: 10.3934/jgm.2014.6.441 |
[9] |
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 |
[10] |
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 |
[11] |
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 |
[12] |
Steven Richardson, Song Wang. The viscosity approximation to the Hamilton-Jacobi-Bellman equation in optimal feedback control: Upper bounds for extended domains. Journal of Industrial and Management Optimization, 2010, 6 (1) : 161-175. doi: 10.3934/jimo.2010.6.161 |
[13] |
Yoshikazu Giga, Przemysław Górka, Piotr Rybka. Nonlocal spatially inhomogeneous Hamilton-Jacobi equation with unusual free boundary. Discrete and Continuous Dynamical Systems, 2010, 26 (2) : 493-519. doi: 10.3934/dcds.2010.26.493 |
[14] |
Nicolas Forcadel, Mamdouh Zaydan. A comparison principle for Hamilton-Jacobi equation with moving in time boundary. Evolution Equations and Control Theory, 2019, 8 (3) : 543-565. doi: 10.3934/eect.2019026 |
[15] |
Yuxiang Li. Stabilization towards the steady state for a viscous Hamilton-Jacobi equation. Communications on Pure and Applied Analysis, 2009, 8 (6) : 1917-1924. doi: 10.3934/cpaa.2009.8.1917 |
[16] |
Alexander Quaas, Andrei Rodríguez. Analysis of the attainment of boundary conditions for a nonlocal diffusive Hamilton-Jacobi equation. Discrete and Continuous Dynamical Systems, 2018, 38 (10) : 5221-5243. doi: 10.3934/dcds.2018231 |
[17] |
Renato Iturriaga, Héctor Sánchez-Morgado. Limit of the infinite horizon discounted Hamilton-Jacobi equation. Discrete and Continuous Dynamical Systems - B, 2011, 15 (3) : 623-635. doi: 10.3934/dcdsb.2011.15.623 |
[18] |
Eddaly Guerra, Héctor Sánchez-Morgado. Vanishing viscosity limits for space-time periodic Hamilton-Jacobi equations. Communications on Pure and Applied Analysis, 2014, 13 (1) : 331-346. doi: 10.3934/cpaa.2014.13.331 |
[19] |
Kai Zhao, Wei Cheng. On the vanishing contact structure for viscosity solutions of contact type Hamilton-Jacobi equations I: Cauchy problem. Discrete and Continuous Dynamical Systems, 2019, 39 (8) : 4345-4358. doi: 10.3934/dcds.2019176 |
[20] |
Ahmad El Hajj, Hassan Ibrahim, Vivian Rizik. $ BV $ solution for a non-linear Hamilton-Jacobi system. Discrete and Continuous Dynamical Systems, 2021, 41 (7) : 3273-3293. doi: 10.3934/dcds.2020405 |
2020 Impact Factor: 2.425
Tools
Metrics
Other articles
by authors
[Back to Top]