HYperbolic and Kinetic Equations :
Asymptotics, Numerics, Analysis
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Short presentation of the team I2

Sites of I2 Ferrara + Trieste (SISSA), Bologna, Brescia, Milano, Pisa, Padova
URL http://dm.unife.it/~I2_to/
Team Organizer L. Pareschi

The team of Ferrara despite the low average age has been doing research in hyperbolic PDE's and kinetic equations since several years. While most of its members are now located at different Italian universities, a large part of them come from the "Institute for Advanced Study of Trieste (SISSA)". In fact, several members of the research groups in Brescia, Milano and Trieste have been former students of A. Bressan at SISSA, during the years 1993-2001.

The research of the group in Trieste (including Udine and Bologna) has been mainly devoted to theoretical aspects of hyperbolic systems of conservation laws, concerning entropy weak solutions for the Cauchy problem and initial-boundary value problems. In the past years, the major contributions include: development of front-tracking algorithms, proofs of uniqueness and stability of weak solutions, analysis of qualitative properties of solutions, examples of finite-time blow-up. Present research is aimed at establishing the uniform stability and the convergence to a unique limit of vanishing viscosity approximations, for hyperbolic systems of conservation laws (task 11 ). This will open up the possibility of a rigorous study of stability and convergence for a wide class of numerical methods (tasks 1 , 15 ). Most collaborations will be with teams I1 and S2.

More applied hyperbolic problems involve members of the research groups in Ferrara, Milano and Brescia. Examples are problems related to traffic flow, phase transition and geometrical optics. In this context (tasks 5 , 7 , 13 ), there exist intense collaboration with teams F2, F3. In particular members of the research groups in Ferrara and Milano have an extensive experience in the development of numerical methods for hyperbolic systems with source terms and collisional kinetic equations. In connection with the SAMI (school for the Applications of Mathematics in Industry) of INDAM (National Institute of High Mathematics - Rome, Italy), the University of Milano-Bicocca has numerous industrial contacts and experience in training for industrial applications. These include simulations of nonlinear fiber optics and semiconductor devices, long range interactions in plasma physics, near continuum computation of rarefied gases and granular media (tasks 1 , 2 , 7 , 10 , 15 , 16 ) . The main collaborations in these directions are with teams A1, I1, I3, D1, F2, F3, S1.

The key scientific staff consists of the following members :

- L. Pareschi (TO) (Ferrara, 30%);  - A. Corli (Ferrara, 25%);  - A. Bressan (SAB) (SISSA-Trieste, 25%);  - F. Ancona (Bologna, 20%);  - P. Baiti (Udine, 20%);  - R. Colombo (Brescia, 25%);  - P. Secchi (Brescia, 20%);  - A. Marson (Brescia, 20%);  - G. Naldi (Milano-Bicocca, 25%);  - G. Guerra (Milano-Bicocca, 20%);  - D. Amadori (Milano, 20%).

The most significant publications for the IHP project are the following:

[1] A. Bressan, Hyperbolic systems of conservation laws. The one dimensional Cauchy problem , Oxford University Press, (2000).

[2] G. Naldi, L. Pareschi, Numerical schemes for hyperbolic systems of conservation laws with stiff diffusive relaxation , SINUM 37 (2000), 1246--1270

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