105 research outputs found
Global smooth solutions of Euler equations for Van der Waals gases
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 ,
for large enough. To do so, we introduce a specific symmetrisation allowing
areas of null density. Next, we make estimates in , using for some terms
the estimates done by M. Grassin, who proved the same theorem in the easier
case of a perfect polytropic gas. We treat the remaining terms separately, due
to their non-linearity
Traffic flow modelling with junctions
AbstractMotivated by the modelling of a roundabout, we are led to study the traffic on a road with points of entry and exit. In this note, we would like to describe the modellisation of a junction and solve the Riemann problem for such a model. More precisely, between each point of discontinuity we use a multi-class extension of the LWR model to describe the evolution of the density of the vehicles, the âmulti-classâ approach being used in order to distinguish the vehicles after their origin and destination. Then, we treat the points of entry and exit thanks to special boundary conditions that give bounds on the flows of the different types of vehicles. In the case of the one-T road we obtain a result of existence and uniqueness. This first step allows us to obtain a similar result for the n-T road. We describe these results and also some properties of the obtained solutions, in order to see how long this model is valid
Existence and uniqueness of measure solutions for a system of continuity equations with non-local flow
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 parameter
Nonlocal Crowd Dynamics Models for several Populations
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 \reali^+. 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 Kruzkov theory of scalar conservation laws in several space
dimensions
Control of the Continuity Equation with a Non Local Flow
This paper focuses on the optimal control of weak (i.e. in general non smooth) 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 a class of equations comprising these models. In particular, we prove the differentiability of solutions with respect to the initial datum and characterize its derivative. A necessary condition for the optimality of suitable integral functionals then follows
Stability and Total Variation Estimates on General Scalar Balance Laws
Consider the general scalar balance law \partial_t u + \Div f(t, x,u) =
F(t,x,u) in several space dimensions. The aim of this note is to estimate the
dependence of its solutions from the flow and from the source . To this
aim, a bound on the total variation in the space variables of the solution is
obtained. This result is then applied to obtain well posedness and stability
estimates for a balance law with a non local source
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