# American Institute of Mathematical Sciences

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 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.

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##### 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.
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).
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).
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).
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).
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).
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).
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).
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).
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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