• Keywords

discontinuous-flux conservation law, Hamilton-Jacobi equation, Highes model, interface coupling conditions

• Profile and skills required

Excellent mastery of Maths contents (Bachelor+Master) in Analysis; good capacity for programing; the interest in fundamental mathematical research motivated by real-life applications.

• Project description

The aim of the thesis is to put into interaction the recent theories of Hamilton-Jacobi equations with discontinuous hamiltonian [2] and of conservation laws with discontinuous flux as well [1].

The first task is to establish the precise connection between the adapted notions of solution and the well-posedness results established for two classes of PDEs with discontinuous non-linearities.

The second task is to develop the theory of the concrete PDE system coupling a Hamilton-Jacobi equation to a conservation law, with the goal to describe pedestrian dynamics

by a variant of mean-field game where every agent follows a trajectory to an exit determined by the global distribution of the pedestrians crowd. This kind of model, formulated for a first time in 2002-03 by Hughes [3], has not yet found an adequate mathematical framework. The goal is to establish, in much generality, existence and stability results for this kind of models and to explore robust strategies of their numerical approximation.

Strating from these two lines of research, other multi-dimensional problems with inner interface can be considered, making appear non-local interactions between agents and strong coupling inducing inner interfaces.

• References

[1] B. Andreianov, K.H. Karlsen, N.H. Risebro. A theory of L1-dissipative solvers for scalar conservation laws with discontinuous flux. Arch. Ration. Mech. Anal. 201 (2011), 27–86

[2]G. Barles, E. Chasseigne. An Illustrated Guide of the Modern Approches of Hamilton-Jacobi Equations and Control Problems with Discontinuities, preprint book (Dec. 2018), https://arxiv.org/abs/1812.09197

[3] R.L. Hughes. A continuum theory for the flow of pedestrians. Transportation Research Part B, 36 (2002), 507–535

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