HYperbolic and Kinetic Equations :
Asymptotics, Numerics, Analysis
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Short presentation of the team A1

Sites of A1 WPI Vienna + Linz, Bayreuth, (ETH Zürich), Praha, Haifa, Tel Aviv
URL http://www.uni-bayreuth.de/departments/math/org/mathe6/staff/memb/grein/hykea1/a1-team.php
Team Organizer G. Rein

The team A1 has been active in the mathematical modeling of physics and engineering problems with partial differential equations, their analysis and numerics for more than 15 years. Main areas of research are: semiconductor modeling, fluid dynamics, mathematical analysis of quantum transport, dispersive equations, homogenization and numerical analysis of Schrödinger type equations, geometric singular perturbation theory, inverse problems, kinetic formulations of PDEs, analysis and numerics of hyperbolic conservation laws, variational techniques, nonlinear diffusion equations, logarithmic Sobolev inequalities, stability of steady states, ...

The team has a strategic role in the network, as its research contribution stands in the cross line of (quantum) kinetic theory and hyperbolic theory. The Linz node is an important center of inverse problems in industrial mathematics with links to a strong"scientific computing center" and long term collaborations with steel industry etc. 
The ETH Zürich node which has valuable experience in numerics of hyperbolic equations, participates without funding.

The team members have extensive experience in running large projects (Wittgenstein prize, two START prize projects) and have been key figures in the previous HCM and TMR network. The Vienna node carries a "doctoral school" on "Differential equations with applications", which imbeds the network training in the national PhD programme. This gives an additional link with the strong Austrian school in biomathematics that is part of the doctoral school.

The team A1 has longstanding research links within the proposed network, in particular with the following network teams : F1 (dispersive equations, nonlinear diffusion), F2 (weak coupling limits), F3 (homogenization, quantum kinetics, Wigner functions), F4 (entropy methods), D1 (open quantum systems), E2 (nonlinear diffusion), I3 (numerical analysis of Schrödinger equations, nonlinear diffusion), S1 (asymptotics of convection-diffusion equations).

The team A1 has done significant work in connection with the following topics contained in the work programme (topic in bold face, non exhaustive )
ad : analysis of various kinetic-to-macroscopic limits for semiconductors and plasmas,
ad : entropy techniques for kinetic and diffusive systems, particularly in developing and adapting the Bakry-Emery approach for nonlinear diffusion-convection problems,
ad 6 , : adaptation of the Wigner formalism to general linear and weakly nonlinear dispersive equations, use of Wigner-function techniques for open quantum systems, analysis of quantum Fokker-Planck equations,
ad 11 : geometric singular perturbation theory,
ad 14 : analysis of (in)compressible fluid models, in particular Navier Stokes equations,
ad 15 : numerical analysis for hyperbolic systems.

The key scientific staff consists of the following members :

- N. J.  Mauser (CO) (WPI, U. Wien, 20%)  - G.  Rein (TO) (WPI, U. Wien, 30%)  - P. A.  Markowich (SC) (WPI, U. Wien 20%)  - C. Schmeiser (WPI, TU-Wien, 20%)  - P. Szmolyan (WPI, TU-Wien, 20%)  - H. Engl (IAB) (U. Linz, 15%)  - W. Burger (U. Linz, 20%)  - R. Jeltsch (ETH Zürich, 15%)  - J. Necas (U. Praha, 20%)  - J. Malek (TC) (U. Praha, 20%)  - M.  Rokyta (U. Praha, 20%)  - E. Feireisl (Academy Praha, 20%)  - S. Necasova (Academy Praha, 20%)  - H. Petzeltova (Academy Praha, 20%)  - G. Wolanski (Technion, Haifa 20%);  - E. Tadmor (SAB) (Tel Aviv U. and UCLA, 15%)

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

[1] P. Gérard (F2), P. A. Markowich (A1), N. J. Mauser (A1), and F. Poupaud (F3), Homogenization limits and Wigner transforms , Comm. Pure Appl. Math., 50 (1997), pp. 323-379.

[2] M. Escobedo (E2), Ph. Laurençot (F3) and E. Feireisl (A1), Large time behavior for degenerate parabolic equations with dominating convective terms , Comm. P.D.E. 25 , no. 1-2 (2000), 73-99

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