Control of the continuity equation with a non local flow

¹ Department of Mathematics, Brescia University, Via Branze 38, 25133 Brescia, Italy. rinaldo@ing.unibs.it
² RWTH Aachen University, Templergraben 55, 52056 Aachen, Germany.
³ Université de Lyon, Université Lyon 1, École Centrale de Lyon, INSA de Lyon, CNRS UMR 5208, Institut Camille Jordan, 43 blvd. du 11 Novembre 1918, 69622 Villeurbanne Cedex, France.

Received: 13 February 2009
Revised: 30 September 2009

Abstract

This paper focuses on the analytical properties of the solutions to the continuity equation with non local flow. Our driving examples are a supply chain model and an equation for the description of pedestrian flows. To this aim, we prove the well posedness of weak entropy solutions in a class of equations comprising these models. Then, under further regularity conditions, we prove the differentiability of solutions with respect to the initial datum and characterize this derivative. A necessary condition for the optimality of suitable integral functionals then follows.