We present a new class of macroscopic models for pedestrian flows. Each individual is assumed to move towards a fixed target, deviating from the best path according to the instantaneous crowd distribution. The resulting equation is a conservation law with a nonlocal flux. Each equation in this class generates a Lipschitz semigroup of solutions and is stable with respect to the functions and parameters defining it. Moreover, key qualitative properties such as the boundedness of the crowd density are proved. Specific models are presented and their qualitative properties are shown through numerical integrations. In particular, the present model accounts for the possibility of reducing the exit time from a room by carefully positioning obstacles that direct the crowd flow.

This paper develops the basic analytical theory related to some recently introduced crowd dynamics models. Where well posedness was known only locally in time, it is here extended to all of R + . The results on the stability with respect to the equations are improved. Moreover, here the case of several populations is considered, obtaining the well posedness of systems of multi-D non-local conservation laws. The basic analytical tools are provided by the classical Kružkov theory of scalar conservation laws in several space dimensions.

Abstract. In this paper, we prove existence and uniqueness of measure solutions for the Cauchy problem associated to the (vectorial) continuity equation with a non-local flow. We also give a stability result with respect to various parameters.Mathematics Subject Classification. 35L65.

We consider a discrete set of individual agents interacting with a continuum. Examples might be a predator facing a huge group of preys, or a few shepherd dogs driving a herd of sheeps. Analytically, these situations can be described through a system of ordinary differential equations coupled with a scalar conservation law in several space dimensions. This paper provides a complete well posedness theory for the resulting Cauchy problem. A few applications are considered in detail and numerical integrations are provided.

Consider the general scalar balance law ∂tu + Divf (t, x, u) = F (t, x, u) in several space dimensions. The aim of this note is to improve the results of Colombo, Mercier, Rosini who gave an estimate of the dependence of the solutions from the flow f and from the source F . The improvements are twofold: first the expression of the coefficients in these estimates are more precise; second, we eliminate some regularity hypotheses thus extending significantly the applicability of our estimates.2000 Mathematics Subject Classification: 35L65.

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We prove global in time existence of solutions of the Euler compressible equations for a Van der Waals gas when the density is small enough in H m , for m large enough. To do so, we introduce a specific symmetrization allowing areas of null density. Next, we make estimates in H m , using for some terms the estimates done by Grassin, who proved the same theorem in the easier case of a perfect polytropic gas. We treat the remaining terms separately, due to their nonlinearity.

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