Rinaldo M. Colombo | University of Brescia (original) (raw)

Papers by Rinaldo M. Colombo

ZAMM, 2007

ZAMM · Z. Angew. Math. Mech. 87, No. 6, 449 – 461 (2007) / DOI 10.1002/zamm.200710327 ... Non loc... more ZAMM · Z. Angew. Math. Mech. 87, No. 6, 449 – 461 (2007) / DOI 10.1002/zamm.200710327 ... Non local balance laws in traffic models and crystal growth ... Rinaldo M. Colombo1,∗ , Andrea Corli2, and Massimiliano D. Rosini1 1 Department of Mathematics, University of Brescia, 25133 Brescia, Italy 2 Department of Mathematics, University of Ferrara, 44100 Ferrara, Italy ... Received 10 January 2007, revised and accepted 18 April 2007 Published online 19 June 2007 ... Key words balance laws, traffic flow models, crystal growth models MSC (2000) 35L50, ...

Journal of Hyperbolic Differential Equations, 2008

Consider n ducts having a common origin and filled with a fluid. Along each duct, the full Euler ... more Consider n ducts having a common origin and filled with a fluid. Along each duct, the full Euler system describes the evolution of the fluid. At the junction, suitable physical conditions couple the n Euler systems. In this paper we prove the well posedness of the Cauchy problem for the model so obtained, provided the total varaiatio of the initial data is sufficiently small.

… and related topics, Jan 1, 2010

In this paper we present the macroscopic model for pedestrian flows proposed by Colombo and Rosin... more In this paper we present the macroscopic model for pedestrian flows proposed by Colombo and Rosini [10] and show its main properties. In particular, this model is able to properly describe the movements of crowds, even after panic has arisen. Furthermore, it is able to reproduce the so called Braess' paradox for pedestrians. From the mathematical point of view, it provides one of the few examples of non classical shocks motivated by real problems, for which a global existence result is available. Finally, its assumptions were experimentally confirmed by an empirical study of a crowd crush on the Jamarat Bridge in Mina, Saudi Arabia, near Mecca, see .

Nonlinear Analysis: Real World Applications, Jan 1, 2009

The main result of this note is the existence of nonclassical solutions to the Cauchy problem for... more The main result of this note is the existence of nonclassical solutions to the Cauchy problem for a scalar conservation law modeling pedestrian flow. From the physical point of view, the main assumption of this model was recently experimentally confirmed in . From the analytical point of view, this model is an example of a conservation law in which nonclassical solutions have a physical motivation and a global existence result for the Cauchy problem with large data is available.

Arxiv preprint arXiv:0810.2462, Jan 1, 2008

Journal of mathematical analysis and …, Jan 1, 2005

Consider the initial -boundary value problem for a Temple system of balance laws. Aim of this pap... more Consider the initial -boundary value problem for a Temple system of balance laws. Aim of this paper is to prove the well posedness of this problem for large times and without requiring the total variation of the initial data be small. 2000 Mathematics Subject Classification: 35L50, 35L60 Key words and phrases: Balance Laws, Initial boundary value problem for conservation laws.

… Problems held in the University of …, Jan 1, 2008

ABSTRACT This presentation is devoted to two macroscopic models for pedestrian traffic. Both are ... more ABSTRACT This presentation is devoted to two macroscopic models for pedestrian traffic. Both are based on scalar conservation laws and aim at the description of a crowd escaping from an area through an exit. The former one is 1D and exploits nonclassical shocks. The latter one uses classical (i.e. weak entropy) solutions in a 2D framework.

ZAMM, 2007

This note addresses the well posedness of Temple systems with non local sources. The resulting th... more This note addresses the well posedness of Temple systems with non local sources. The resulting theorem holds globally in time and without requiring any smallness of the initial data. Its scope comprises models for traffic flow and for crystal growth. 2000 Mathematics Subject Classification: 35L50, 90B20, 82D25.

Annali della Scuola normale superiore di Pisa. Classe di scienze, 1998

RefDoc Refdoc est un service / is powered by. ...

This work presents a new model for the movement, the erosion and the deposition of granular matte... more This work presents a new model for the movement, the erosion and the deposition of granular matter along a sloping bed. It is a synthesis of the Hadeler-Kuttler and of the Savage-Hutter models. The result is a 3×3 system of balance laws able to describe the deposition-erosion dynamics, as in the former model, while being compliant with the dynamics, as in the latter one. First, the basic analytical properties of the new model are described. Then several numerical simulations allow to compare the different models. Whenever the slope of the bed changes, sign and deposition-erosion phenomena are relevant, the present model appears to provide better descriptions of granular matter behaviour.

We consider systems of conservation laws with nonlocal sources, ie,∂ tu+∂ xf (u)= G (u),(1) where... more We consider systems of conservation laws with nonlocal sources, ie,∂ tu+∂ xf (u)= G (u),(1) where f is the flow of a nonlinear hyperbolic system of conservation laws and G: L1↦→ L1 is a (possibly) nonlocal operator. As examples, we consider below the case G (u)= g (u)+ Q∗ u that enters a classical radiating gas model, see [22], as well as Rosenau regularization of Chapman–Enskog expansion of the Boltzmann equation, see [19, 20]. Here, by nonlocal we mean nonlocal in the space variable. Related results concerning the time variable, ie, ...

This paper presents a model for 2 inviscid, immiscible, compressible and isentropic fluids in 1 s... more This paper presents a model for 2 inviscid, immiscible, compressible and isentropic fluids in 1 space dimension. Its well posedness is proved, globally in time, for data having small total variation. In a sample non-smooth case, the limit in which one of the fluids becomes incompressible is characterized.

This paper is devoted to general balance laws (with a possibly non-local source term) with a non-... more This paper is devoted to general balance laws (with a possibly non-local source term) with a non-characteristic boundary. Basic well posedness results are obtained. New uniqueness results for the solutions to conservation and/or balance laws with boundary are also provided.

Abstract Consider an n× n system of hyperbolic balance laws with coinciding shock and rarefaction... more Abstract Consider an n× n system of hyperbolic balance laws with coinciding shock and rarefaction curves. This note proves the well posedness in the large of this system, provided there exists a domain that is invariant both with respect to the homogeneous conservation law and to the ordinary differential system generated by the right hand side. No “non-resonance” hypothesis is assumed.

The theory of hyperbolic conservation laws has been successfully applied to the study of vehicula... more The theory of hyperbolic conservation laws has been successfully applied to the study of vehicular traffic flows. We present here some models showing phase transitions, that in terms of traffic flows correspond to two distinct behaviors, free or congested.

This note is devoted to the explicit construction of a functional defined on all pairs of L1 func... more This note is devoted to the explicit construction of a functional defined on all pairs of L1 functions with small total variation, which is equivalent to the L1 distance and non-increasing along the trajectories of a given system of conservation laws. Two different constructions are provided, yielding an extension of the original stability functional of Bressan, Liu and Yang.

Abstract We consider a discrete set of individual agents interacting with a continuum. Examples m... more Abstract 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 sheep. 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 p-system describing the subsonic flow of a fluid in a pipe with section a= a (x). We... more Consider the p-system describing the subsonic flow of a fluid in a pipe with section a= a (x). We prove that the resulting Cauchy problem generates a Lipschitz semigroup, provided the total variation of the initial datum and the oscillation of a are small. An explicit estimate on the bound of the total variation of a is provided, showing that at lower fluid speeds, higher total variations of a are acceptable. An example shows that the bound on TV (a) is mandatory, for otherwise the total variation of the solution may grow arbitrarily.

ZAMM, 2007

ZAMM · Z. Angew. Math. Mech. 87, No. 6, 449 – 461 (2007) / DOI 10.1002/zamm.200710327 ... Non loc... more ZAMM · Z. Angew. Math. Mech. 87, No. 6, 449 – 461 (2007) / DOI 10.1002/zamm.200710327 ... Non local balance laws in traffic models and crystal growth ... Rinaldo M. Colombo1,∗ , Andrea Corli2, and Massimiliano D. Rosini1 1 Department of Mathematics, University of Brescia, 25133 Brescia, Italy 2 Department of Mathematics, University of Ferrara, 44100 Ferrara, Italy ... Received 10 January 2007, revised and accepted 18 April 2007 Published online 19 June 2007 ... Key words balance laws, traffic flow models, crystal growth models MSC (2000) 35L50, ...

Journal of Hyperbolic Differential Equations, 2008

Consider n ducts having a common origin and filled with a fluid. Along each duct, the full Euler ... more Consider n ducts having a common origin and filled with a fluid. Along each duct, the full Euler system describes the evolution of the fluid. At the junction, suitable physical conditions couple the n Euler systems. In this paper we prove the well posedness of the Cauchy problem for the model so obtained, provided the total varaiatio of the initial data is sufficiently small.

… and related topics, Jan 1, 2010

In this paper we present the macroscopic model for pedestrian flows proposed by Colombo and Rosin... more In this paper we present the macroscopic model for pedestrian flows proposed by Colombo and Rosini [10] and show its main properties. In particular, this model is able to properly describe the movements of crowds, even after panic has arisen. Furthermore, it is able to reproduce the so called Braess' paradox for pedestrians. From the mathematical point of view, it provides one of the few examples of non classical shocks motivated by real problems, for which a global existence result is available. Finally, its assumptions were experimentally confirmed by an empirical study of a crowd crush on the Jamarat Bridge in Mina, Saudi Arabia, near Mecca, see .

Nonlinear Analysis: Real World Applications, Jan 1, 2009

The main result of this note is the existence of nonclassical solutions to the Cauchy problem for... more The main result of this note is the existence of nonclassical solutions to the Cauchy problem for a scalar conservation law modeling pedestrian flow. From the physical point of view, the main assumption of this model was recently experimentally confirmed in . From the analytical point of view, this model is an example of a conservation law in which nonclassical solutions have a physical motivation and a global existence result for the Cauchy problem with large data is available.

Arxiv preprint arXiv:0810.2462, Jan 1, 2008

Journal of mathematical analysis and …, Jan 1, 2005

Consider the initial -boundary value problem for a Temple system of balance laws. Aim of this pap... more Consider the initial -boundary value problem for a Temple system of balance laws. Aim of this paper is to prove the well posedness of this problem for large times and without requiring the total variation of the initial data be small. 2000 Mathematics Subject Classification: 35L50, 35L60 Key words and phrases: Balance Laws, Initial boundary value problem for conservation laws.

… Problems held in the University of …, Jan 1, 2008

ABSTRACT This presentation is devoted to two macroscopic models for pedestrian traffic. Both are ... more ABSTRACT This presentation is devoted to two macroscopic models for pedestrian traffic. Both are based on scalar conservation laws and aim at the description of a crowd escaping from an area through an exit. The former one is 1D and exploits nonclassical shocks. The latter one uses classical (i.e. weak entropy) solutions in a 2D framework.

ZAMM, 2007

This note addresses the well posedness of Temple systems with non local sources. The resulting th... more This note addresses the well posedness of Temple systems with non local sources. The resulting theorem holds globally in time and without requiring any smallness of the initial data. Its scope comprises models for traffic flow and for crystal growth. 2000 Mathematics Subject Classification: 35L50, 90B20, 82D25.

Annali della Scuola normale superiore di Pisa. Classe di scienze, 1998

RefDoc Refdoc est un service / is powered by. ...

This work presents a new model for the movement, the erosion and the deposition of granular matte... more This work presents a new model for the movement, the erosion and the deposition of granular matter along a sloping bed. It is a synthesis of the Hadeler-Kuttler and of the Savage-Hutter models. The result is a 3×3 system of balance laws able to describe the deposition-erosion dynamics, as in the former model, while being compliant with the dynamics, as in the latter one. First, the basic analytical properties of the new model are described. Then several numerical simulations allow to compare the different models. Whenever the slope of the bed changes, sign and deposition-erosion phenomena are relevant, the present model appears to provide better descriptions of granular matter behaviour.

We consider systems of conservation laws with nonlocal sources, ie,∂ tu+∂ xf (u)= G (u),(1) where... more We consider systems of conservation laws with nonlocal sources, ie,∂ tu+∂ xf (u)= G (u),(1) where f is the flow of a nonlinear hyperbolic system of conservation laws and G: L1↦→ L1 is a (possibly) nonlocal operator. As examples, we consider below the case G (u)= g (u)+ Q∗ u that enters a classical radiating gas model, see [22], as well as Rosenau regularization of Chapman–Enskog expansion of the Boltzmann equation, see [19, 20]. Here, by nonlocal we mean nonlocal in the space variable. Related results concerning the time variable, ie, ...

This paper presents a model for 2 inviscid, immiscible, compressible and isentropic fluids in 1 s... more This paper presents a model for 2 inviscid, immiscible, compressible and isentropic fluids in 1 space dimension. Its well posedness is proved, globally in time, for data having small total variation. In a sample non-smooth case, the limit in which one of the fluids becomes incompressible is characterized.

This paper is devoted to general balance laws (with a possibly non-local source term) with a non-... more This paper is devoted to general balance laws (with a possibly non-local source term) with a non-characteristic boundary. Basic well posedness results are obtained. New uniqueness results for the solutions to conservation and/or balance laws with boundary are also provided.

Abstract Consider an n× n system of hyperbolic balance laws with coinciding shock and rarefaction... more Abstract Consider an n× n system of hyperbolic balance laws with coinciding shock and rarefaction curves. This note proves the well posedness in the large of this system, provided there exists a domain that is invariant both with respect to the homogeneous conservation law and to the ordinary differential system generated by the right hand side. No “non-resonance” hypothesis is assumed.

The theory of hyperbolic conservation laws has been successfully applied to the study of vehicula... more The theory of hyperbolic conservation laws has been successfully applied to the study of vehicular traffic flows. We present here some models showing phase transitions, that in terms of traffic flows correspond to two distinct behaviors, free or congested.

This note is devoted to the explicit construction of a functional defined on all pairs of L1 func... more This note is devoted to the explicit construction of a functional defined on all pairs of L1 functions with small total variation, which is equivalent to the L1 distance and non-increasing along the trajectories of a given system of conservation laws. Two different constructions are provided, yielding an extension of the original stability functional of Bressan, Liu and Yang.

Abstract We consider a discrete set of individual agents interacting with a continuum. Examples m... more Abstract 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 sheep. 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 p-system describing the subsonic flow of a fluid in a pipe with section a= a (x). We... more Consider the p-system describing the subsonic flow of a fluid in a pipe with section a= a (x). We prove that the resulting Cauchy problem generates a Lipschitz semigroup, provided the total variation of the initial datum and the oscillation of a are small. An explicit estimate on the bound of the total variation of a is provided, showing that at lower fluid speeds, higher total variations of a are acceptable. An example shows that the bound on TV (a) is mandatory, for otherwise the total variation of the solution may grow arbitrarily.