A flame propagation model on a network with application to a blocking problem

Fabio Camilli; Elisabetta Carlini; Claudio Marchi

doi:10.3934/dcdss.2018051

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

October 2018, 11(5): 825-843. doi: 10.3934/dcdss.2018051

A flame propagation model on a network with application to a blocking problem

Fabio Camilli ^1,, , Elisabetta Carlini ^2, and Claudio Marchi ^3,

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

* Corresponding author

Received February 2017 Revised August 2017 Published June 2018

We consider the Cauchy problem

$\left\{ \begin{array}{*{35}{l}} {{\partial }_{t}}u+H(x,Du) = 0&(x,t)\in \Gamma \times (0,T) \\ u(x,0) = {{u}_{0}}(x)&x\in \Gamma \\\end{array} \right.$

where

$\Gamma$

is a network and

$H$

is a positive homogeneous Hamiltonian which may change from edge to edge. In the first part of the paper, we prove that the Hopf-Lax type formula gives the (unique) viscosity solution of the problem. In the latter part of the paper we study a flame propagation model in a network and an optimal strategy to block a fire breaking up in some part of a pipeline; some numerical simulations are provided.

Keywords: Evolutive Hamilton-Jacobi equation, viscosity solution, network, Hopf-Lax formula, approximation.

Mathematics Subject Classification: Primary: 35D40; Secondary: 35R02, 35F21, 65M06, 49L25.

Citation: Fabio Camilli, Elisabetta Carlini, Claudio Marchi. A flame propagation model on a network with application to a blocking problem. Discrete and Continuous Dynamical Systems - S, 2018, 11 (5) : 825-843. doi: 10.3934/dcdss.2018051

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.

Figure 1. Test1. Graph structure where $R_0$ is represented by the circle marker and the vertices by the rhombus markers (Top Left). Color map of the time $u_h(x)$ at which a node $x$ get burnt, computed by (29), (Top Right) and its 3D view (Bottom).

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2. Test1. Time to reach a point $x$ from the operation center $x_0$ (circle marker) and set of the admissible nodes $V^h_{ad}$ (square marker). 2D view (Left) and 3D view (Right).

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3. Test1. Optimal blocking strategy $\sigma ^h_{opt}$ (square marker), preserved network region (cross marker) and minimum burnt network region (continuum line) starting from $R_0$ (circle marker).

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4. Test2. Graph structure where $R_0$ is represented by the circle markers and the vertices by the rhombus markers (Top Left). Color map of the time $u_h(x)$ at which a node $x$ get burnt, computed by (29) (Top Right), and its 3D view (Bottom).

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 5. Test2. Time to reach a point $x$ from the operation center $x_0$ (circle marker) and set of the admissible nodes $V^h_{ad}$ (square markers). 2D view(Left) and 3D view (Right).

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 6. Test2. Optimal blocking strategy $\sigma ^h_{opt}$ (square markers), preserved network region (thin line) and minimum burnt network region (thick line) starting from $R_0$ (circle markers).

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 7. Test3. Graph structure where $R_0$ is represented by the circle markers and the vertices by the rhombus markers (Top Left). Color map of the time $u_h(x)$ at which a node $x$ get burnt, computed by (29), (Top Right) and its 3D view (Bottom).

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 8. Test3. Time to reach a point $x$ from the operation center $x_0$ (circle marker) and set of the admissible nodes $V^h_{ad}$ (square markers). 2D view(Left) and 3D view (Right).

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 9. Test3. Optimal blocking strategy $\sigma ^h_{opt}$ (square marker), preserved network region (thin line) and minimum burnt network region (thick line) starting from $R_0$ (circle marker).

Figure Options

Download full-size image

Download as PowerPoint slide

[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

Tools

Article outline

Figures and Tables

2020 Impact Factor: 2.425

Tools

Figures and Tables

2020 Impact Factor: 2.425

American Institute of Mathematical Sciences

A flame propagation model on a network with application to a blocking problem

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Tools

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

A flame propagation model on a network with application to a blocking problem

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Tools

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors