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

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

* 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.
Citation: Fabio Camilli, Elisabetta Carlini, Claudio Marchi. A flame propagation model on a network with application to a blocking problem. Discrete & Continuous Dynamical Systems - S, 2018, 11 (5) : 825-843. doi: 10.3934/dcdss.2018051
References:
[1]

Y. AchdouF. CamilliA. 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.  Google Scholar

[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.  Google Scholar

[3]

G. Barles, Remark on a Flame Propagation Model, Report INRIA #464, 1985. Google Scholar

[4]

G. BarlesH. M. Soner and P. Souganidis, Front propagations and phase field theory, SIAM J. Control Optim., 31 (1993), 439-469.  doi: 10.1137/0331021.  Google Scholar

[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.  Google Scholar

[6]

F. CamilliC. 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.  Google Scholar

[7]

F. CamilliA. 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.  Google Scholar

[8]

Y. G. ChenY. 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.  Google Scholar

[9]

G. CostesequeJ.-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.  Google Scholar

[10]

M. Garavello and B. Piccoli, Traffic Flow on Networks, AIMS Series on Applied Mathematics, American Institute of Mathematical Sciences, Springfield, MO, 2006.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

D. Mugnolo, Semigroup Methods for Evolution Equations on Networks, Understanding Complex Systems, Springer, Berlin, 2014. doi: 10.1007/978-3-319-04621-1.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

P. Van MieghemJ. Omic and R. Kooij, Virus spread in Networks, IEEE/ACM Trans. on networking, 17 (2009), 1-14.  doi: 10.1109/TNET.2008.925623.  Google Scholar

show all references

References:
[1]

Y. AchdouF. CamilliA. 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.  Google Scholar

[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.  Google Scholar

[3]

G. Barles, Remark on a Flame Propagation Model, Report INRIA #464, 1985. Google Scholar

[4]

G. BarlesH. M. Soner and P. Souganidis, Front propagations and phase field theory, SIAM J. Control Optim., 31 (1993), 439-469.  doi: 10.1137/0331021.  Google Scholar

[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.  Google Scholar

[6]

F. CamilliC. 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.  Google Scholar

[7]

F. CamilliA. 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.  Google Scholar

[8]

Y. G. ChenY. 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.  Google Scholar

[9]

G. CostesequeJ.-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.  Google Scholar

[10]

M. Garavello and B. Piccoli, Traffic Flow on Networks, AIMS Series on Applied Mathematics, American Institute of Mathematical Sciences, Springfield, MO, 2006.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

D. Mugnolo, Semigroup Methods for Evolution Equations on Networks, Understanding Complex Systems, Springer, Berlin, 2014. doi: 10.1007/978-3-319-04621-1.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

P. Van MieghemJ. Omic and R. Kooij, Virus spread in Networks, IEEE/ACM Trans. on networking, 17 (2009), 1-14.  doi: 10.1109/TNET.2008.925623.  Google Scholar

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 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 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 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 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 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 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 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 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).
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