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Elena ROSSI

Professore Associato
Dipartimento di Scienze e Metodi dell'Ingegneria

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Pubblicazioni

2024 - NON LINEAR HYPERBOLIC–PARABOLIC SYSTEMS WITH DIRICHLET BOUNDARY CONDITIONS [Articolo su rivista]
Colombo, Rinaldo M.; Rossi, Elena
abstract

We prove the well posedness of a class of non linear and non local mixed hyperbolic–parabolic systems in bounded domains, with Dirichlet boundary conditions. In view of control problems, stability estimates on the dependence of solutions on data and parameters are also provided. These equations appear in models devoted to population dynamics or to epidemiology, for instance.

2023 - Analysis of district heating networks [Articolo su rivista]
Borsche, Raul; Eimer, Matthias; Garavello, Mauro; Rossi, Elena
abstract

In the context of district heating networks we consider a model for the distribution of energy through water (for heating or cooling) from a central power station to the consumers. We prove the well posedness of the system, by using the Banach Fixed Point Theorem together with stability estimates for reduced systems. Eventually we consider optimal control problems motivated by applications and we provide the existence of optimal controls in special situations.

2023 - General renewal equations motivated by biology and epidemiology [Articolo su rivista]
Colombo, R. M.; Garavello, M.; Marcellini, F.; Rossi, E.
abstract

We present a unified framework ensuring well posedness and providing stability estimates to a class of Initial – Boundary Value Problems for renewal equations comprising a variety of biological or epidemiological models. This versatility is achieved considering fairly general – possibly non linear and/or non local – interaction terms, allowing both low regularity assumptions and independent variables with or without a boundary. In particular, these results also apply, for instance, to a model for the spreading of a Covid like pandemic or other epidemics. Further applications are shown to be covered by the present setting.

2022 - Vaccination strategies through intra—compartmental dynamics [Articolo su rivista]
Colombo, Rinaldo M.; Marcellini, Francesca; Rossi, Elena
abstract

We present a new epidemic model highlighting the roles of the immunization time and concurrent use of different vaccines in a vaccination campaign. To this aim, we introduce new intra-compartmental dynamics, a procedure that can be extended to various other situations, as detailed through specific case studies considered herein, where the dynamics within compartments are present and influence the whole evolution.

2021 - Nonlocal approaches for multilane traffic models [Articolo su rivista]
Friedrich, Jan; Göttlich, Simone; Rossi, Elena
abstract

We present a multilane traffic model based on balance laws, where the nonlocal source term is used to describe the lane changing rate. The modelling framework includes the consideration of local and nonlocal flux functions. Based on a Godunov-type numerical scheme, we provide BV estimates and a discrete entropy inequality. Together with the L1-contractivity property, we prove existence and uniqueness of weak solutions. Numerical examples show the nonlocal impact compared to local flux functions and local sources.

2021 - Well-posedness and control in a hyperbolic–parabolic parasitoid–parasite system [Articolo su rivista]
Colombo, R. M.; Rossi, E.
abstract

We develop a time and space-dependent predator—prey model. The predators' equation is a nonlocal hyperbolic balance law, while the diffusion of prey obeys a parabolic equation, so that predators “hunt” for prey, while prey diffuse. A control term allows to describe the use of predators as parasitoids to limit the growth of prey–parasites. The general well-posedness and stability results here obtained ensure the existence of optimal pest control strategies, as discussed through some numerical integrations. The specific example we have in mind is that of Trichopria drosophilæ used to fight against the spreading of Drosophila suzukii.

2020 - A modeling framework for biological pest control [Articolo su rivista]
Colombo, Rm; Rossi, E
abstract

We present an analytic framework where biological pest control can be simulated. Control is enforced through the choice of a time and space dependent function representing the deployment of a species of predators that feed on pests. A sample of different strategies aimed at reducing the presence of pests is considered, evaluated and compared. The strategies explicitly taken into account range, for instance, from the uniform deployment of predators on all the available area over a short/long time interval, to the alternated insertion of predators in different specific regions, to the release of predators in suitably selected regions. The effect of each strategy is measured through a suitably defined cost, essentially representing the total amount of prey present over a given time interval over all the considered region, but the variation in time of the total amount of pests is also evaluated. The analytic framework is provided by an integro-differential hyperbolic-parabolic system of partial differential equations. While prey diffuse according to the usual Laplace operator, predators hunt for prey, moving at finite speed towards regions of higher prey density.

2020 - An age and space structured SIR model describing the Covid-19 pandemic [Articolo su rivista]
Colombo, R. M.; Garavello, M.; Marcellini, F.; Rossi, E.
abstract

We present an epidemic model capable of describing key features of the Covid-19 pandemic. While capturing several qualitative properties of the virus spreading, it allows to compute the basic reproduction number, the number of deaths due to the virus and various other statistics. Numerical integrations are used to illustrate the adherence of the evolutions described by the model to specific well known real features of the present pandemic. In particular, this model is consistent with the well known relevance of quarantine, shows the dramatic role of care houses and accounts for the increase in the death toll when spatial movements are not constrained.

2020 - Comparative study of macroscopic traffic flow models at road junctions [Articolo su rivista]
Goatin, P; Rossi, E
abstract

We qualitatively compare the solutions of a multilane model with those produced by the classical Lighthill-Whitham-Richards equation with suitable coupling conditions at simple road junctions. The numerical simulations are based on the Godunov and upwind schemes. Several tests illustrate the models' behaviour in different realistic situations.

2020 - Well-posedness of a non-local model for material flow on conveyor belts [Articolo su rivista]
Rossi, E.; Weissen, J.; Goatin, P.; Gottlich, S.
abstract

In this paper, we focus on finite volume approximation schemes to solve a non-local material flow model in two space dimensions. Based on the numerical discretisation with dimensional splitting, we prove the convergence of the approximate solutions, where the main difficulty arises in the treatment of the discontinuity occurring in the flux function. In particular, we compare a Roe-type scheme to the well-established Lax-Friedrichs method and provide a numerical study highlighting the benefits of the Roe discretisation. Besides, we also prove the L1-Lipschitz continuous dependence on the initial datum, ensuring the uniqueness of the solution.

2019 - A multilane macroscopic traffic flow model for simple networks [Articolo su rivista]
Goatin, P; Rossi, E
abstract

We prove the well-posedness of a system of balance laws inspired by [H. Holden and N. H. Risebro, SIAM J. Math. Anal., 51 (2019), pp. 3694-3713], describing macroscopically the traffic flow on a multilane road network. Motivated by real applications, we allow us for the the presence of space discontinuities both in the speed law and in the number of lanes. This allows us to describe a number of realistic situations. Existence of solutions follows from compactness results on a sequence of Godunov's approximations, while L\bfone -stability is obtained by the doubling of variables technique. Some numerical simulations illustrate the behavior of solutions in sample cases.

2019 - Modelling crowd movements in domains with boundaries [Articolo su rivista]
Colombo, Rm; Rossi, E
abstract

This paper contributes to the macroscopic modeling of crowd movements. The presented model is non local, i.e., it takes into account interactions among pedestrians at different distances. Particular care is given to how non local interactions are influenced by walls, obstacles and exits. The resulting dynamics captures various well known patterns of crowd movements, such as the clogging of exits and the spontaneous formation of queues. The careful choice of obstacles near an exit is shown to be able to reduce evacuation times. An ad hoc numerical algorithm is detailed, some of its properties discussed and the convergence of the corresponding approximate solutions is investigated.

2019 - Stability estimates for non-local scalar conservation laws [Articolo su rivista]
Chiarello, Fa; Goatin, P; Rossi, E
abstract

We prove the stability of entropy weak solutions of a class of scalar conservation laws with non-local flux arising in traffic modelling. We obtain an estimate of the dependence of the solution with respect to the kernel function, the speed and the initial datum. Stability is obtained from the entropy condition through doubling of variable technique. We finally provide some numerical simulations illustrating the dependencies above for some cost functionals derived from traffic flow applications.

2019 - Stability of the 1D IBVP for a non autonomous scalar conservation law [Articolo su rivista]
Colombo, Rm; Rossi, E
abstract

We prove the stability with respect to the flux of solutions to initial-boundary value problems for scalar non autonomous conservation laws in one space dimension. Key estimates are obtained through a careful construction of the solutions

2019 - Well-posedness of general 1d initial boundary value problems for scalar balance laws [Articolo su rivista]
Rossi, Elena
abstract

We focus on the initial boundary value problem for a general scalar balance law in one space dimension. Under rather general assumptions on the flux and source functions, we prove the well-posedness of this problem and the stability of its solutions with respect to variations in the flux and in the source terms. For both results, the initial and boundary data are required to be bounded functions with bounded total variation. The existence of solutions is obtained from the convergence of a Lax-Friedrichs type algorithm with operator splitting. The stability result follows from an application of Kruzkov's doubling of variables technique, together with a careful treatment of the boundary terms.

2019 - Well‐posedness of IBVP for 1D scalar non‐local conservation laws [Articolo su rivista]
Goatin, P; Rossi, E
abstract

We consider the initial boundary value problem (IBVP) for a non‐local scalar conservation law in one space dimension. The non‐local operator in the flux function is not a mere convolution product, but it is assumed to be aware of boundaries. Introducing an adapted Lax‐Friedrichs algorithm, we provide various estimates on the approximate solutions that allow to prove the existence of solutions to the original IBVP. The uniqueness follows from the Lipschitz continuous dependence on initial and boundary data, which is proved exploiting results available for the local IBVP

2018 - Definitions of solutions to the ibvp for multi-dimensional scalar balance laws [Articolo su rivista]
Rossi, Elena
abstract

We consider four definitions of solution to the initial-boundary value problem (IBVP) for a scalar balance laws in several space dimensions. These definitions are extended to the same most general framework and then compared. The first aim of this paper is to detail differences and analogies among them. We focus then on the ways the boundary conditions are fulfilled according to each definition, providing also connections among these various modes. The main result is the proof of the equivalence among the presented definitions of solution.

2018 - IBVPs for scalar conservation laws with time discontinuous fluxes [Articolo su rivista]
Colombo, Rm; Rossi, E
abstract

The initial boundary value problem for a class of scalar nonautonomous conservation laws in 1 space dimension is proved to be well posed and stable with respect to variations in the flux. Targeting applications to traffic, the regularity assumptions on the flow are extended to a merely L ∞ dependence on time. These results ensure, for instance, the well-posedness of a class of vehicular traffic models with time-dependent speed limits. A traffic management problem is then shown to admit an optimal solution.

2018 - Nonlocal Conservation Laws in Bounded Domains [Articolo su rivista]
Colombo, Rm; Rossi, E
abstract

The well posedness for a class of nonlocal systems of conservation laws in a bounded domain is proved and various stability estimates are provided. This construction is motivated by the modeling of crowd dynamics, which also leads to defining a nonlocal operator adapted to the presence of a boundary. Numerical integrations show that the resulting model provides qualitatively reasonable solutions.

2016 - A mixed hyperbolic-parabolic system to describe predator-prey dynamics [Relazione in Atti di Convegno]
Rossi, Elena
abstract

Following [2], a model aiming at the description of two competing populations is introduced. In particular, it is considered a nonlinear system consisting of a nonlocal conservation law for predators coupled with a parabolic equation for prey. The drift term in the equation for predators is in general a nonlocal and nonlinear function of the prey density: the movement of predators can hence be directed towards regions where the concentration of prey is higher. Lotka-Volterra type right hand sides describe the feeding. In [2] the resulting Cauchy problemis proved to be well posed in any space dimension with respect to the L1 topology, and estimates on the growth of the solution in L1 and L∞norm and on the time dependence are provided. Numerical integrations show a few qualitative features of the solutions. This is a joint work with RinaldoM. Colombo.

2016 - Biological and industrial models motivating nonlocal conservation laws: A review of analytic and numerical results [Articolo su rivista]
Colombo, Rm; Marcellini, Francesca; Rossi, Elena
abstract

This paper is devoted to the overview of recent results concerning nonlocal systems of conservation laws. First, we present a predator - prey model and, second, a model for the laser cutting of metals. In both cases, these equations lead to interesting pattern formation.

2016 - Convergence of a numerical scheme for a mixed hyperbolic-parabolic system in two space dimensions [Articolo su rivista]
Rossi, Elena; Schleper, V.
abstract

We prove the convergence of an explicit numerical scheme for the discretization of a coupled hyperbolic-parabolic system in two space dimensions. The hyperbolic part is solved by a Lax-Friedrichs method with dimensional splitting, while the parabolic part is approximated by an explicit finite-difference method. For both equations, the source terms are treated by operator splitting. To prove convergence of the scheme, we show strong convergence of the hyperbolic variable, while convergence of the parabolic part is obtained only weakly∗ in L∞. The proof relies on the fact that the hyperbolic flux depends on the parabolic variable through a convolution function. The paper also includes numerical examples that document the theoretically proved convergence and display the characteristic behaviour of the Lotka-Volterra equations.

2016 - Non local mixed systems and IBVPs for balance laws [Abstract in Atti di Convegno]
Rossi, Elena
abstract

2015 - Erratum to Hyperbolic predators vs. parabolic prey [Commun. Math. Sci., 13, 2, (2015) 369-400] [Articolo su rivista]
Colombo, Rm; Rossi, E
abstract

We correct an error in the proof of the main result in [R.M. Colombo and E. Rossi, Commun. Math. Sci., 13(2):369-400, 2015]. The theorem, with all the provided estimates, remains true. The online version is corrected.

2015 - Hyperbolic predators vs. parabolic prey [Articolo su rivista]
Colombo, Rm; Rossi, E
abstract

We present a nonlinear predator-prey system consisting of a nonlocal conservation law for predators coupled with a parabolic equation for prey. The drift term in the predators' equation is a nonlocal function of the prey density, so that the movement of the predators can be directed towards regions with high prey density. Moreover, Lotka-Volterra type right hand sides describe the feeding. A theorem ensuring existence, uniqueness, continuous dependence of weak solutions, and various stability estimates is proved, in any space dimension. Numerical integrations show a few qualitative features of the solutions.

2015 - Rigorous estimates on balance laws in bounded domains [Articolo su rivista]
Colombo, Rm; Rossi, Elena
abstract

The initial-boundary value problem for a general balance law in a bounded domain is proved to be well posed. Indeed, we show the existence of an entropy solution, its uniqueness and its Lipschitz continuity as a function of time, of the initial datum and of the boundary datum. The proof follows the general lines in [4], striving to provide a rigorous treatment and detailed references.

2014 - A justification of a LWR model based on a follow the leader description [Articolo su rivista]
Rossi, E
abstract

We investigate the correlations between a macroscopic Lighthill-Whitham and Richards model and a microscopic follow-the-leader model for traffic flow. We prove that the microscopic model tends to the macroscopic one in a sort of kinetic limit, i.e. as the number of individuals tends to infinity, keeping the total mass fixed. Based on this convergence result, we approximately compute the solutions to a conservation law by means of the integration of an ordinary difierential system. From the numerical point of view, the limiting procedure is then extended to the case of several populations, referring to the macroscopic model in [2] and to the natural multi-population analogue of the microscopic one.

2014 - On the micro-macro limit in traffic flow [Articolo su rivista]
Colombo, R; Rossi, Elena
abstract

We investigate the relations between a macroscopic Lighthill-Whitham and Richards model and a microscopic follow-the-leader model for traffic flow. Solutions to the microscopic model are proved to tend to those to the macroscopic one in a sort of kinetic limit, i.e. as the number of individuals tends to +∞ while their total mass is constant. Based on this convergence result, we approximately compute the solutions to a conservation law by means of the integration of an ordinary differential system.

2013 - A rigorous result on the discrete–continuum limit in traffic flow [Abstract in Atti di Convegno]
Rossi, E
abstract