## Séminaire d'Analyse

**Balance laws in domains with boundaries: from definitions of solution to applications**

Elena Rossi

mardi 05 juin 2018 14:00 - Tours - Salle 1180 (Bât E2)

**Résumé :**

This talk is focused on initial boundary value problems (IBVPs) for balance laws. We begin by presenting various definitions of solution to the problem that are found in the literature, highlighting differences and analogies among them. In particular, the way the boundary conditions are satisfied according to each definition plays a relevant role and constitutes the key ingredient in the proof of the equivalence among the considered definitions. We recap the well-posedness results found in the literature, with a particular attention to stability estimates with respect to the flux. Although this estimate is still missing for the general IBVP, we are able to recover it in some special frameworks, namely for 1D non autonomous conservation laws and for linear multiD conservation laws, though limited to homogeneous boundary conditions. This latter result allows to prove the well-posedness of a class of system of non local conservation laws in bounded domains. This construction is motivated by the modelling of crowd dynamics. One of the novelty lies in the introduction of a non local operator aware of boundaries. We conclude the talk by presenting some numerical integrations, which show that the model provides qualitatively reasonable solutions.

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