1b. Project objectives

One of the major general objectives of this project is to strengthen the European collaboration between hyperbolic and kinetic communities in a set-up where people working in modeling, analysis, numerics, and applications cooperate. There is a remarkable complementarity in the work style and methods of the different places in Europe represented by the fifteen teams, which is optimally exploited by the proposed network.

By this (unique by international standards) proposed project, which collects a number of strong scientists that are fluent in diverse methodologies, the chances for"breakthroughs" increase. To this end, the inclusion of younger scientist around the PhD which is the core of this"research training network" is a strategic asset. There are indeed extremely hard problems among the objectives where any progress could be considered a breakthrough.

The research objectives of the project are results of open discussions among many scientists of the fifteen groups; already the setup and definition of the programme is a significant achievement and the full "collaboration database" of possible research objectives of these community is a strategic asset which in some form will be part of the useful information to be disseminated in the course of this network.

Primarily responsible for the choice of research objectives is the Scientist in Charge (SiC), Benoit Perthame, who is assisted by the Scientific Advisory Board (SAB)
The SAB is a council of senior scientists, where each group sends one representative, resulting in a complementary, broad, and experienced advisory board. It cooperates with the SiC, who is the"head" by definition, in order to make strategic scientific decisions, in particular for a scientific reassessment before the midterm review.

The members of the SAB are E. Tadmor (A1), M. Esteban (F1), B. Perthame (F2), F. Poupaud (F3), D. Serre (F4), H. Spohn (D1), S. Müller (D2), T. Souganidis (G1), A. Marquina (E1), J.L. Vazquez (E2), M. Pulvirenti (I1), A. Bressan (I2), G. Toscani (I3), L. Arkeryd (S1), B. Engquist (S2).

A formal list of the project objectives, grouped into sixteen tasks, can be found in the"work programme" in part C.4. The objectives/tasks are ordered according to the following four main themes:

• A. Multiscale phenomena and limiting structures
• B. Collisional evolution equations
• C. Nonlinear waves and hyperbolic systems
• D. Numerical methods and simulations

Below we list and explain the major goals of the project, as far as predictable at present. This list is not just the sum of what the individual scientists are currently doing; indeed, several applications of this kind of hyperbolic or kinetic equations where members of this network have collaborations and expertise have been omitted, such as quantum chemistry, biomathematics, or financial mathematics. However, it is a particular strength of this network that it can immediately react to new developments and deliver the most timely research training in an important, application driven field of mathematics.

The objectives and work program are deliberately not focussed on one or two topics, but are kept rather broad; this should prove beneficial for delivering a training with complementary and interdisciplinary aspects. These objectives are not only supposed to spark progress of science as such, but are supposed to serve the goal of high-level training for internationally mobile researchers who can also switch between academia and industry.

Exemplary list of "objectives" (O) and "breakthroughs to expect" (B):

Analysis

(B) Zero viscosity limits for systems of conservation laws
(O/B) Viscosity limits with physical diffusion matrices
(B) Limit from Boltzmann to incompressible Navier-Stokes and Euler
(O/B) Entropy methods : Wasserstein metrics in PDEs
(O/B) Classical and semiclassical limits via Wigner measures; defocusing/short-time-focusing NLS
(O) (Asymptotic) Analysis of the of Dirac-Maxwell equation
(O) Generalization of the use of multiresolution methods for hyperbolic equations
(O) Improving the knowledge of averaging lemmas in order to study regularity in hyperbolic conservation laws and some Ginzburg-Landau functional

Modeling and analysis

(O/B) Understanding coagulation-fragmentation, also by numerical simulations
(O/B) Quantum Boltzmann equation
(O/B) Granular media : hydrodynamic models; rapid granular flows
(O) Viscosity methods for shock waves in multi-dimensional systems
(O) Convergence of BGK expansions and other relaxation approximations to fluid equations
(O) Boundary layer analysis; nonlinear stability of phase boundaries

Numerics and analysis

(O)] Stability and convergence of finite difference schemes for hyperbolic problems and fluid models
(O)] Viscosity, Godunov schemes for hyperbolic numerics
(O)] Development of efficient numerical algorithms for balance laws with source terms.
(O)] Generalisation of multiresolution schemes
(O)] Efficient and robust numerical methods for the Boltzmann equation and other kinetic models
(O)] Nonlinear Schrödinger equations and nonlinear Vlasov equations; numerical techniques for Schrödinger equations in semiclassical regimes
(O)] Control theory and optimization in hyperbolic and kinetic equations