Flux-limited solutions for quasi-convex Hamilton-Jacobi equations on networks

By C. Imbert, Régis Monneau. 2013

111 citations recorded by the source corpus.

We study Hamilton-Jacobi equations on networks in the case where Hamiltonians are quasi-convex with respect to the gradient variable and can be discontinuous with respect to the space variable at vertices. First, we prove that imposing a general vertex condition is equivalent to imposing a specific one which only depends on Hamiltonians and an additional free parameter, the flux limiter. Second, a general method for proving comparison principles is introduced. This method consists in constructing a vertex test function to be used in the doubling variable approach. With such a theory and such a method in hand, we present various applications, among which a very general existence and uniqueness result for quasi-convex Hamilton-Jacobi equations on networks.

Subfields: Computational Theory and Mathematics, Computer Networks and Communications, Control and Systems Engineering

View Paper: https://www.semanticscholar.org/paper/af62b400c9b80d2ac38631c889920a40b1675484

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In Computational Theory and Mathematics

In Computer Networks and Communications

In Control and Systems Engineering

Citations in the corpus

Viscosity solutions for junctions: well posedness and stability (2016, methodology): “The following definition was introduced in [5].”
Mean field games with state constraints: from mild to pointwise solutions of the PDE system (2018, result): “Such a rigidity result is reminiscent of the reformulation of the notion of viscosity solution of Hamilton-Jacobi equation with state-constraints in terms of fluxlimited solutions, as described in the recent papers by Imbert and Monneau [25] and Guerand [21].”