Rinaldo M. Colombo

Hyperbolic Phase Transitions in Traffic Flow

This paper provides a mathematical model of the phenomenon of phase transitions in traffic flow. The model consists of a scalar conservation law coupled with a 2 × 2 system of conservation laws. The coupling is achieved via a free boundary, where the phase transition takes place. For this model, the Riemann problem is stated and globally solved. The Cauchy problem is proved to admit a solution defined globally in time without any assumption about the smallness of the initial data or the number of phase boundaries. Qualitative properties of real traffic flow are shown to agree with properties of the solutions of the model.

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We present an

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-populations model for traffic flow

-population generalization of the Lighthill–Whitham and Richards traffic flow model. This model is analytically interesting because of several non-standard features. For instance, it leads to non-classical shocks and enjoys an unexpected stability in spite of the presence of umbilic points. Furthermore, while satisfying all the minimal ‘common sense’ requirements, it also allows for a description of phenomena often neglected by other models, such as overtaking.

The semigroup generated by 2 × 2 conservation laws

Consider the Cauchy problem for a strictly hyperbolic 2×2 system of conservation laws in one space dimension: {ie1-01} assuming that each characteristic field is either linearly degenerate or genuinely nonlinear. This paper develops a new algorithm, based on wave-front tracking, which yields a Cauchy sequence of approximate solutions, converging to a unique limit depending continuously on the initial data. The solutions that we obtain constitute a semigroup S, defined on a set {ie1-02} of integrable functions with small total variation. For some Lipschitz constant L, we have the estimate {ie1-03}

A class of nonlocal models for pedestrian traffic

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.

A 2 × 2 hyperbolic traffic flow model

A new traffic flow model is presented. It accounts for various qualitative features of the evolution of the density and speed of cars along a crowded road. The model consists of a 2 x 2 system of nonlinear hyperbolic conservation laws generating a Cauchy problem which is well posed for all reasonable initil data. A similar result can be proved for the initial boundary value problem. The presence of a speed limit is also considered.

A Well Posed Riemann Problem for the

This work is devoted to the solution to Riemann Problems for the

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--System at a Junction

-system at a junction, the main goal being the extension to the case of an ideal junction of the classical results that hold in the standard case.