4. Work plan

Research objectives and tasks

The proposal objectives are divided into the following sixteen tasks which are grouped under four main themes; please note that several of the tasks actually relate to more than one of the main themes. In particular, many of the tasks under A, B, and C have a strong numerical component, which is in accordance with our integrated modeling-analysis-numerics approach.

A. Multiscale phenomena and limiting structures

1. Kinetic and relaxation methods for conservation laws and moment systems
(Derivation and validation of models using diffusive and hydrodynamic limits, ...)
2. Multiscale phenomena for charged particle systems
(Modeling, analysis and numerics of hybrid semiconductor and plasma models, ...)
3. Mass transportation and entropy methods
(Large-time asymptotics for kinetic and parabolic equations, ...)
4. Eulerian-Lagrangian methods
(Multiphase flows, particles and solids in fluids and sprays, ...)
5. Continuum and kinetic models of phase transitions
(Coagulation/fragmentation, Bose condensation, kinetic relations, ...)
6. Quantum evolution equations
(Hartree-Fock equation, Dirac equation, semiclassical/nonrelativistic limits, ...)

B. Collisional evolution equations

7. Transport and collisions
(Analysis of and numerics for the Boltzmann equation, traffic flow problems, ...)
8. Many-particle systems
(Asymptotic limits, derivation of: kinetic equations, fluid models, and nonlinear one-particle approximations, ...)
9. Collision and dispersive quantum models
(Derivation of "quantum Boltzmann" equations, electron-phonon interaction, weak coupling limits, interaction with random media, ...)
10. Kinetic models for granular media and materials
(Derivation, analysis and numerics for models with inelastic collisions and/or mean fields, thin film magnetism, ...)

C. Nonlinear waves and hyperbolic systems

11. Structural properties of hyperbolic systems
(Zero-viscosity limits, diffusive and dispersive effects, non-strict hyperbolicity, ...)
12. Stability of nonlinear waves and boundary layers
(Over-/undercompressible shock phenomena, discrete shock profiles, ...)
13. Hamilton-Jacobi equations and geometrical optics
(Thin film magnetism, mesh generation, pressureless gas, ...)
14. Fluid dynamics equations
(Navier-Stokes equation, rotating flows, atmospheric and oceanic flows, dynamo equations, ...)

D. Numerical methods and simulations

15. Numerical methods for balance laws and dispersive equations
(Central schemes, relaxation schemes, kinetic schemes, schemes for nonlinear Schrödinger eq., ...)
16. Computational singular (in)compressible flows
(Magneto-hydrodynamics, multiphase models, shallow water equation, semiconductor simulations, Euler equations, Navier-Stokes equations, ...)

Distribution of tasks, responsibilities, and collaboration in the network

Table 1 shows which of the teams work on or are responsible for the tasks listed above. Here, a cross "X" means that the team is working on that task; the sign "O" means that the team is responsible for that task.

Table 1 Distribution of tasks and responsibilities.

Teams
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16

A1
X
X
X
O
X
X
O
X

F1
X
X
X
O
X
X
X
X
X
X
X
X

F2
X
X
X
X
X
X
X
X
O
X
X
X

F3
X
O
X
X
X
X
X
O
X
X
X
X
X

F4
X
X
O
X
X
X
X
X
X
X
X
X

D1
X
X
X
X
O
X
X
X
X
X

D2
X
X
X
X
X
X
X
X

G1
X
X
X
O
X
X

I1
O
X
O
X
X
X
X
X
X
X
X
X

I2
X
X
X
X
X
O
X
X
X

I3
X
X
X
X
X
O
X

E1
X
X
X
X
X
X

E2
X
X
X
O

S1
X
O
X
X
X
X
X
X

S2
X
X
X
X
X
O

Table 2 shows the intended collaborations among the teams; note the additional graph on an extra page. It should be stressed that many of these collaborations are already operational.

Collaborating teams will periodically organize small-scale workshops where the work in progress will be monitored with respect to the work schedule and milestones described below, and where round-table discussions will spark new ideas. Of course, the work will continue in the network teams and benefit from contacts with the collaborating teams.

Table 3 indicates the professional effort on the network tasks of each team.

Resources of the network

Clearly, the main resource of the network is the extensive research and training expertise of the participants, documented in Section C.5. Moreover, most of the network teams are hosting other research and/or training programs, funded by national or European institutions and in part documented in Section C.5. These resources will lend additional impetus to the proposed network, and the latter will in turn greatly increase the resulting research and training value of these additional resources.

Table 2 : Collaboration of the teams and responsible groups ( see also enclosed graphic)

Teams
A1
F1
F2
F3
F4
D1
D2
G1
I1
I2
I3
E1
E2
S1
S2

A1
X
X
X
X
X
X
X
X
X
X
X
X

F1
X
X
X
X
X
X
X
X
X
X

F2
X
X
X
X
X
X
X
X
X
X
X
X

F3
X
X
X
X
X
X
X
X
X
X
X
X

F4
X
X
X
X
X
X
X
X
X

D1
X
X
X
X
X
X
X
X

D2
X
X
X
X
X
X
X
X
X
X

G1
X
X
X
X
X
X
X
X
X

I1
X
X
X
X
X
X
X
X
X
X

I2
X
X
X
X
X
X
X
X
X
X
X

I3
X
X
X
X
X
X
X
X
X
X
X
X
X

E1
X
X
X
X
X
X
X

E2
X
X
X
X
X
X
X
X
X

S1
X
X
X
X
X
X
X
X
X
X
X

S2
X
X
X
X
X
X
X
X
X

Table 3 : Professional research effort on the network project

Group
Young researchers to be financed by the contract (person-months) (a)
Researchers to be financed from other sources (person-months) (b)
Researchers likely to contribute to the project (number of individuals) (c)

A1
48
112
15

F1
36
146
16

F2
38
99
12

F3
40
158
19

F4
36
122
9

D1
27,5
97
12

D2
60
110
13

G1
42
90
10

I1
34
149
17

I2
49
90
11

I3
37
88
12

E1
44
120
13

E2
37
139
14

S1
30
153
17

S2
25
83
11

TOTALS
583,5
1756
201

Work schedule and milestones

In Table 4 we give a schedule of the most important goals (milestones) concerning the tasks listed above. The Scientist in Charge together with the Scientific Advisory Board, cf. Section C.1 b, will keep the research efforts on track with this work schedule in mind.

Annual conferences on a fairly large scale, usually lasting a full week, will put in perspective the research accomplishments of the network and formulate appropriate future goals. These meetings will serve both a scientific and a training purpose: graduate students are encouraged to attend, and young researchers are prominent among the speakers.

The first annual conference in spring/summer 2002 in Vienna is already in planning, with cofinancing of the local projects of the team A1, in particular the ESI programme of L.A. Caffarelli and P.A. Markowich.

Of course, the whole conference and workshop activity of the HYKE community will be coordinated; in fact, this network is a major motivation to optimize the activities in this respect, at least on the European level.

Limitations to planning

Progress in mathematics, i. e. new ideas leading to mathematical breakthroughs, cannot be planned for. What we have planned for is a scientific set-up, on the levels of research objectives, group interaction, organization, and schedules, which will stimulate as much as possible the creation of excellent new mathematics. There is a considerable flexibility built into our network which will make possible timely reaction to new promising trends, optimal exploitation of the hoped-for breakthroughs, and redirection of efforts in case some research objective should prove less fruitful. What can be planned for to a larger extent is a close-to-optimal research training situation for young scientists, which we are trying to implement.

Project topic group
Midterm
Third year

A: Multiscale phenomena and limiting structures Large-time asymptotics using entropy methods; analysis of hybrid semiconductor models; analysis of Hartree-Fock type models. Analysis of Dirac equations; kinetic relations to phase transition problems

Coagulation-fragmentation: Description of the gelation profile and new numerical methods for computing the gelation time applications to combustion

application of Sobolev inequalities to large
- time asyptotics of non
- homogeneous kinetic equations

B: Collisional evolution equations Analysis of Boltzmann equation and traffic flow problems; derivation and analysis of "quantum Boltzmann" equations Derivation of granular media models incorporating inelastic collisions

C: Nonlinear waves and hyperbolic systems Zero-viscosity limit of hyperbolic systems; stability of nonlinear waves Analysis of Hamilton
- Jacobi equations; derivation and analysis of rotating flow models

D: Numerical methods and simulations Relaxation schemes for balance laws Semiconductor simulations for hybrid models

Convergence of Godunov scheme (small BV solutions for general hyperbolic systems) Existence for partial viscosity (small BV solutions for general hyperbolic systems)

Study of multiresolution schemes for stiff problems comparison with finite volume methods

Analysis of spectral discretisation of nonlinear Schrödinger equations

Group	Young researchers to be financed by the contract (person-months) (a)	Researchers to be financed from other sources (person-months) (b)	Researchers likely to contribute to the project (number of individuals) (c)
A1	48	112	15
F1	36	146	16
F2	38	99	12
F3	40	158	19
F4	36	122	9
D1	27,5	97	12
D2	60	110	13
G1	42	90	10
I1	34	149	17
I2	49	90	11
I3	37	88	12
E1	44	120	13
E2	37	139	14
S1	30	153	17
S2	25	83	11
TOTALS	583,5	1756	201

Project topic group	Midterm	Third year
A: Multiscale phenomena and limiting structures	Large-time asymptotics using entropy methods; analysis of hybrid semiconductor models; analysis of Hartree-Fock type models.	Analysis of Dirac equations; kinetic relations to phase transition problems
	Coagulation-fragmentation: Description of the gelation profile and new numerical methods for computing the gelation time	applications to combustion
		application of Sobolev inequalities to large - time asyptotics of non - homogeneous kinetic equations
B: Collisional evolution equations	Analysis of Boltzmann equation and traffic flow problems; derivation and analysis of "quantum Boltzmann" equations	Derivation of granular media models incorporating inelastic collisions
C: Nonlinear waves and hyperbolic systems	Zero-viscosity limit of hyperbolic systems; stability of nonlinear waves	Analysis of Hamilton - Jacobi equations; derivation and analysis of rotating flow models
D: Numerical methods and simulations	Relaxation schemes for balance laws	Semiconductor simulations for hybrid models
	Convergence of Godunov scheme (small BV solutions for general hyperbolic systems)	Existence for partial viscosity (small BV solutions for general hyperbolic systems)
	Study of multiresolution schemes for stiff problems	comparison with finite volume methods
	Analysis of spectral discretisation of nonlinear Schrödinger equations