# Program

**Monday, 10th**

15:20 -- 15:30 Welcome and Opening

15:30 -- 16:30 Main speaker

16:30 -- 17:00 Invited speaker

17:00 -- 17:30 Invited speaker

17:30 -- 18:00 Invited speaker

**Tuesday, 11th**

10:00 -- 11:00 Main speaker

11:00 -- 11:30 Invited speaker

11:30 -- 12:00 Invited speaker

12:00 -- 12:30 Invited speaker

*Lunch break*

15:30 -- 16:30 Main speaker

16:30 -- 17:00 Invited speaker

17:00 -- 17:30 Invited speaker

17:30 -- 18:00 Invited speaker

**Wednesday, 12th**

09:30 -- 10:30 Main speaker

10:30 -- 11:00 Invited speaker

11:00 -- 11:30 Invited speaker

11:30 -- 12:00 Invited speaker

12:00 -- 12:30 Invited speaker

### Abstracts

**Giacomo Albi** - Multistep methods in hyperbolic systems with relaxation

We are interested in high-order linear multistep schemes for hyperbolic systems with multi-scale relaxation. These systems has the property of being either hyperbolic or parabolic, according to the choice of certain parameter. Hence we will propose a general (IMEX) multistep technique, which is able to cope with the loose of efficiency in stiff regimes, and able to capture the asymptotic limit. Several numerical examples will confirm the theoretical results. Finally, we will show how to use part of these techniques in the context of optimal control problems, showing some applications in conservation laws.

**Debora Amadori** - A semilinear hyperbolic system with space-dependent and nonlinear damping

We consider a semilinear hyperbolic system with space-dependent and nonlinear damping. We present an analysis of the large time behavior that stems from a probabilistic interpretation of the equation. As a result, accurate information on their asymptotic behavior and exponential decay towards stationary solutions are obtained. Joint work with Fatima Aqel and Edda Dal Santo (University of L'Aquila).

**Roberta Bianchini** - On a vector-BGK approximation of fluid-dynamics equations

We consider a singular semilinear hyperbolic approximation to the Euler and the incompressible Navier-Stokes equations in 2D, inspired by the kinetic theory. We show the convergence of the vector-BGK model to the Euler equations under the hyperbolic scaling, and to the incompressible Navier-Stokes equations in the diffusive scaling.

**Felisia Angela Chiarello** - Nonlocal multiclass traffic flow models

In this talk, I will consider the framework of the non-local traffic flow models. I will prove the existence in finite time of weak solutions for a class of nonlocal systems in one space dimension, approximating the problem by a Godunov type numerical scheme. Finally, I will show some numerical simulations. Joint work with Paola Goatin from Inria Sophia Antipolis - Mediterranee.

**Maria Teresa Chiri** - Conservation law models for supply chains on a network with finite buffers

We introduce a new model for supply chains on a network based on conservation laws with discontinuous flux evolving on each arc and on buffers of limited capacity in every junction. The flux is discontinuous at the maximal density (of processed parts) since it admits different values according with the free or congested status of the supply chain. We establish the well-posedness of the Cauchy problem with bounded and integrable initial data. The key ingredient is the analysis of discontinuous Hamilton-Jacobi equations associated to the conservation laws evolving on each arc. This is a joint work with Fabio Ancona (Universita' di Padova).

**Rinaldo M. Colombo** - Balance Laws in Population Dynamics

Different aspects of the dynamics of various sorts of populations can be described through balance laws. This presentation overviews some of them, hopefully underlining those that are of more interest to the audience.

**Marco Di Francesco** - Rigorous derivation of scalar conservation laws via deterministic particles

We provide an overview of results on the derivation of scalar conservation laws via ODEs systems of deterministic particles interacting via follow-the-leader interactions. The main motivation behind this problem arises in traffic flow modelling. We present results on the derivation of entropy solutions of the Cauchy problem of the LWR model and later extensions of this result on problems with Dirichlet boundary data and on similar models such as the ARZ model for traffic flow and the Hughes model for pedestrians. The results are joint with S. Fagioli, M. D. Rosini and G. Russo.

**Raul De Maio** - Some open problems for multiscale models on networks

In the first part of my talk I will shortly introduce the transport equation for measures on networks. This approach can be adapted to micro or macro scales and it allows us to model and describe many phenomena in vehicular traffic, such as interactions between drivers, stop&go waves and congestions. In the second part, I will focus on some open problems and applications which arise quite naturally from the multiscale models and which could be interesting research ideas on vehicular traffic or transport on networks.

**Simone Fagioli** - System of nonlocal interactions PDEs with Newtonian potentials in 1d

We deal with a nonlocal interaction system of two species in dimension one, with Newtonian kernels. Depending on the initial data, different approaches are studied in order to tackle the existence and uniqueness problem. Existing results using the method of characteristics break down due to the non-Lipschitz characteristic velocity fields. Furthermore, we highlight an interesting link to hyperbolic systems in non-conservative form.

**Raffaele Folino** - Slow dynamics for conservation laws with saturating diffusion.

In this talk, I will discuss some recent results on the long-time dynamics of the solutions to an IBVP for a Burgers-type equation with a saturating nonlinear diffusion. In particular, I will focus the attention on the metastable dynamics of the solutions, by analyzing the differences with the classic viscous Burgers equation.

**Elio Marconi** - Structure and regularity of entropy solutions to scalar conservation laws

We consider scalar conservation laws in one space dimension without convexity assumptions on the flux. The notion of Lagrangian representation will be introduced for bounded entropy solutions in order to investigate the structure of their characteristics. This also allows to deduce a priori regularity estimates in terms of spaces of functions with generalized bounded variation.

**Stefano Modena** - Non-uniqueness for the transport equation with Sobolev vector fields

We construct a large class of examples of non-uniqueness for the linear transport equation and the transport-diffusion equation with divergence-free vector fields belonging to Sobolev spaces. Our result can be seen as a counterpart to DiPerna and Lions’ well-posedness theorem (joint with L. Székelyhidi).

**Sabrina Francesca Pellegrino** - Representation of capacity drop at a road merge via point constraints in a first order traffic model

We reproduce the capacity drop phenomenon at a road merge via point constraints at the junction in a first order traffic model. We first construct an enhanced version of the locally constrained model introduced by Haut, Bastin and Chitour in [1], then we propose its counterpart featuring a non-local constraint and finally we compare numerically the two models by constructing an adapted finite volumes scheme. The results are joint with C. Donadello, E. Dal Santo and M. D. Rosini.

References

[1] Haut, B., Bastin, G., and Chitour, Y. A macroscopic traffic model for road networks with a representation of the capacity drop phenomenon at the junctions. In Proceedings 16th IFAC World Congress, Prague, Czech Republic (2005).

**Gabriella Puppo** - Unstable waves in kinetic traffic models

Kinetic traffic models can be thought of as a relaxation system towards a conservation law, where the relaxation velocities can be interpreted as the microscopic velocities of the single cars, and the conservation law states that the number of cars in conserved. In this interpretation, the microscopic velocities are non negative (you can’t drive backward in a lane) and with this restriction standard asymptotic expansions exhibit instabilities. The problem becomes: are these instabilities really bad news?

**Emanuela Radici** - Deterministic particle approximation for scalar aggregation-diffusion equations with nonlinear mobility

We investigate the existence of weak type solutions of a class of aggregation-diffusion PDEs with nonlinear mobility obtained as large particle limit of a suitable nonlocal version of the follow-the-leader scheme, which is interpreted as the discrete Lagrangian approximation of the target continuity equation. This is a joint work with Marco Di Francesco and Simone Fagioli.

**Marta Strani** - The role of a regularization in hyperbolic instabilities

I will discuss the influence of a regularizing term in the instabilities of some hyperbolic problems; in particular I will show that, for a class of hyperbolic equations that are strongly unstable, a small regularizing term introduces a time-delay in the aforementioned instability. This is ultimately related to a change in the behavior of the equation under consideration, which presents a transition from hyperbolicity to ellipticity. First, I will discuss a viscous regularization of the complex Burgers equation. Finally, I will mention what happens when regularizing the Euler equations with a dispersive term. Some of these results have been obtained in collaboration with B. Texier (Université Paris Diderot).

**Giuseppe Visconti** - Qualitative Properties of a Continuous Model for Data Flow in Large Computer Systems

In this talk, we propose a conditioned hyperbolic system of PDEs modeling data flow in large computer systems. We analyze weak solutions and use the relaxation system associated to the model in order to employ an implicit-explicit scheme and, consequently, to numerically investigate the solutions of the model. Finally, we analyze a control problem where the objective is the minimization of the processing time. The optimal control is then applied on the distribution of the processing rate among the processors. Some of these results have been obtained in collaboration with C. Hauck (CAM Group, Oak Ridge National Laboratory, Oak Ridge, USA) and M. Herty (IGPM, RWTH Aachen University, Aachen, Germany)

References

[1] Barnard, Hauck - A Hydrodynamical Model of Data Flow - Tech. Report. Oak Ridge National Laboratory (2017)

[2] Hauck, Herty, Visconti - Qualitative Properties of a Continuous Model for Data Flow in Large Computer Systems - In preparation (2018)