A flame propagation model on a network with application to a blocking problem
For researchers in PDEs on networks and fire safety engineering, this work extends theoretical results to networks and offers a practical application, though the contribution is incremental.
The paper proves a Hopf-Lax formula for viscosity solutions of a Hamilton-Jacobi equation on a network and applies it to a flame propagation model, providing an optimal blocking strategy for fire in a pipeline with numerical simulations.
We consider the Cauchy problem \[\partial_t u+H(x,Du)=0 \quad (x,t)\inΓ\times (0,T),\quad u(x,0)=u_0(x) \quad x\inΓ\] where $Γ$ is a network and $H$ is a convex and positive homogeneous Hamiltonian which may change from edge to edge. In the former 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.