Nicola Bellomo Mario Pulvirenti

Editors

Modeling in Applied Sciences A Kinetic Theory Approach

Birkhäuser Boston • Basel • Berlin

CONTENTS

Preface xiii

Chapter 1. General ized Kinet i c M o d e l s in Appl i ed Sc iences 1 by N. Bellomo and M. Pulvirenti

1.1 Introduction 1 1.2 The Boltzmann Equation 3 1.3 The Vlasov or Mean-Field Equation 10 1.4 Generalized Kinetic Models 11 1.5 Generalized Models and Plan of the Book . . . 13 1.6 References 18

Chapter 2. R a p i d Granulär Flows: Kinet i c s and H y d r o d y n a m i c s 21 by I. Goldhirsch

2.1 Introduction 21 2.2 One Dimensional Hydrodynamics 24

^ 2.2.1 -Introductory remarks 24 2.2.2 The System 25 2.2.3 Homogeneous dynamics: Mean field results 25 2.2.4 Hydrodynamic equations 27

2.3 The Two Dimensional Case: Stationary Shear Flow 31

2.3.1 Introduction 31 2.3.2 Formulation of the problem 32 2.3.3 Perturbative expansion 33 2.3.4 The first order term 35 2.3.5 The second order term 37 2.3.6 The stress tensor 39 2.3.7 Summary of Section 2.3 41

2.4 The Unsteady Two Dimensional Case 42 2.4.1 Introduction 42 2.4.2 Formulation of the problem 42

v

vi Kinetic Models in Applied Sciences

2.5 The Three Dimensional Case: Hydrodynamic Equations 48

2.5.1 Introduction 48 2.5.2 Formulation of the problem 49 2.5.3 Method of Solution 50 2.5.4 Solution at 0{K) 52 2.5.5 Solution at 0(e) 55 2.5.6 Solution at ö(Ke) 56 2.5.7 Contribution of the Ö(K2) terms . . . . 57 2.5.8 Constitutive relations 59

2.6 Boundary Conditions 61 2.6.1 The elastic case 65 2.6.2 Solubility conditions and some results . . 68 2.6.3 The inelastic case' 70

2.7 Conclusions, Problems and Outlook 72 2.8 References 74

Chapter 3 . CoUect ive Behavior of One-Dimens iona l Granulär M e d i a 81 by D. Benedetto, E. Caglioti, and M. Pulvirenti

3.1 Introduction 81 3.2 The Microscopic Model 83 3.3 Collapses 84 3.4 The Quasielastic Limit 87 3.5 The Mean-Field Equation 90 3.6 The Hydrodynamic Behavior of the

Mean Field Equation 92 3.7 One-Dimensional Boltzmann Equation . . . . 97 3.8 Heating the System 100 3.9 A Hydrodynamical Picture 103

3.10 The Diffusive Limit 107 3.11 References 108

Chapter 4. N o t e s on Mathemat i ca l P r o b l e m s on t h e D y n a m i c s of Dispersed Part ic les Interact ing through a Fluid 111 by P.E. Jabin and B. Perthame

4.1 Introduction 111 4.2 Dynamics of Balls in a Potential Flow . . . . 1 1 5

4.2.1 The füll dynamics 116 4.2.2 The method of reflections 118

Contents vii

4.2.3 The dipole approximation 119 4.3 Kinetic Theory for the Hamiltonian System of

Bubbly Flows 121 4.3.1 The general Lagrangian structure . . . . 121 4.3.2 The corresponding Hamiltonian structure 122 4.3.3 The mean field equation 123

4.4 Numerical Simulation in the Case of a Potential Flow and Short Range Effect 126

4.5 Interaction of Particles in a Stokes Flow . . . . 129 4.5.1 Notations 130 4.5.2 Case of a Single bubble and Stokeslets . . 131 4.5.3 The method of reflections 133 4.5.4 The dipole approximation 133

4.6 Kinetic and Macroscopic Equations for Particles in a Stokes Flow 135

4.6.1 The general interaction model 135 4.6.2 Energy and long time behavior for the

kinetic equation 137 4.6.3 A macroscopic equation 138

4.7 Numerical Simulations for Stokes Flow . . . . 139 4.7.1 Introduction 139 4.7.2 Presentation of the computation . . . . 143 4.7.3 Conclusions 144

4.8 References 145

Chapter 5. T h e Becker—Döring Equat ions 149 by M. Slemrod

5.1 Introduction 149 5.2 Existence of Solutions to the Becker-Döring

Equations 151 5.3 Trend to Equilibrium 155 5.4 Metastable States 159 5.5 Large Time Asymptotic Revised:

Lifschitz-Slyozov and Wagner Evolution . . . 163 5.6 References 170

Chapter 6. Nonl inear Kinet i c M o d e l s w i t h Chemica l Reac t ions 173 by C.P. Grünfeld

6.1 Introduction 173 6.2 Boltzmann Equations for Reacting Gas . . . . 178

viii Kinetic Models in Applied Sciences

6.2.1 Extended kinetic theory with creation and removal 181

6.2.2 Generalized Boltzmann equations . . . . 183 6.3 General Properties of Solutions 192

6.3.1 The initial value problem 192 6.3.2 The H-theorem, equilibrium properties

and mass action law 197 6.3.3 Outlines of proofs 202

6.4 Analytical Solutions, Approximation Methods, Reactive Fluid Dynamic Limits 207

6.4.1 Analytical Solutions 208 6.4.2 Approximation methods 210 6.4.3 Reactive fluid dynamic limits 214

6.5 Concluding Remarks and Open Problems . . . 219 6.6 References 221

Chapter 7. D e v e l o p m e n t of B o l t z m a n n M o d e l s in M a t h e m a t i c a l B io logy 225 by N. Bellomo and S. Stöcker

7.1 Introduction 225 7.2 The Boltzmann Equation in

Population Dynamics 227 7.3 A Few Notes on the Cauchy Problem 233 7.4 Application in Mathematical

Epidemiology 234 7.5 Application in Mathematical Immunology . . . 239 7.6 A Survey of Applications 248 7.7 Developments and Perspectives 254

7.7.1 Models with internal structure 254 7.7.2 Models with time structure 255 7.7.3 Research perspectives on modeling . . . 256 7.7.4 Research perspectives on

analytic topics 257 7.8 The Interplay between Mathematics and

Immunology 257 7.9 References 259

Chapter 8. Kine t i c Trafflc Flow M o d e l s 263 by A. Klar and R. Wegener

8.1 Introduction 263 8.2 Basic Concepts 264

Contents ix

8.2.1 Levels of descriptions and notations . . . 264 8.2.2 Homogeneous traffic flow 266

8.3 Microscopic Models 270 8.3.1 Car following modeis 270 8.3.2 A multilane microscopic model 272 8.3.3 Cellular automata modeis 276

8.4 Kinetic Models 276 8.4.1 The Prigogine model 277 8.4.2 The Paveri-Fontana model 278 8.4.3 Boltzmann versus Enskog type kinetic

modeis 279 8.4.4 A kinetic multilane model 280

8.5 Macroscopic Models 289 8.5.1 Basic modeis 289 8.5.2 Models with an acceleration

equation 290 8.5.3 A derived fluid dynamic model 292

8.6 Numerical Simulations 299 8.6.1 Simulation of the microscopic model . . . 299 8.6.2 Simulation of the cumulative homogeneous

kinetic model and computation of macroscopic coefficients 303

8.6.3 Inhomogeneous simulations 307 8.7 References 313

Chapter 9. Kinet ic Limits for Large C o m m u n i c a t i o n Networks 317 by C. Grahm

9.1 Introduction 317 9.1.1 The scope of this document 319 9.1.2 Kinetic limits for chaotic initial laws

and in equilibrium 319 9.1.3 Development of this document 320

9.2 Examples of Networks and of Related Practical Issues 321

9.2.1 Invariant laws, and the Erlag fixed point approximation 321

9.2.2 A star-shaped loss network 322 9.2.3 A queuing network with selection of the

shortest.among several queues 324 9.2.4 A fully-connected loss network with

alternative routing 325

x Kinetic Models in Applied Sciences

9.3 Preliminaries 326 9.3.1 General notation and terminology . . . . 326 9.3.2 The Skorohod space 327 9.3.3 General network notation 328 9.3.4 Chaoticity, exhangeability, and laws of

large numbers 328 9.4 Mean-Field Networks and Propagation

of Chaos 330 9.4.1 Mean-field modeis and nonlinear limits . . 330 9.4.2 Propagation of chaos 333

9.5 Chaoticity in Equilibrium 335 9.6 Chaoticity for the Star-Shaped

Loss Network •. 338 9.6.1 Martingale formulations, and equations

for the marginals 339 9.6.2 Propagation of chaos 341 9.6.3 Chaoticity in equilibrium 342

9.7 Chaoticity for the Queuing Network with Selection of the Shortest among Several Queues 342

9.7.1 Martingale formulations, and equations for the marginals 343

9.7.2 Propagation of chaos 344 9.7.3 Chaoticity in equilibrium 346

9.8 Propagation of Chaos Using Random Graphs andTrees 348

9.8.1 The fully connected loss network with alternative routing 348

9.8.2 Propagation of chaos for a general class of networks 351

9.8.3 The chaos hypothesis and the empirical measures 353

9.8.4 The limit Boltzmann tree and Boltzmann processes 357

9.8.5 Propagation of chaos under slight symmetry assumptions 358

9.9 Functional Central Limit and Large Deviation Results 359

9.9.1 Central limit theorems 359 9.9.2 Large deviation results 366

9.10 Conclusions and Perspectives 366 9.10.1 Propagation of chaos 366 9.10.2 Chaoticity in equilibrium 367

Contents xi

9.10.3 Central limit and large deviation results 367

9.11 References 368

Chapter 10. Numer ica l S imulat ion of t h e B o l t z m a n n Equat ion by Part ic le M e t h o d s 371

by J. Struckmaier

10.1 Introduction 371 10.2 Particle Methods for the Boltzmann

Equation 373 10.2.1 Approximation of functions

by particles 374 10.2.2 Spatial-homogeneous Boltzmann equation 378 10.2.3 Spatial-inhomogeneous problems . . . . 383 10.2.4 Generalized time Integration schemes . . 387 10.2.5 Extensions to steady-state problems . . . 392 10.2.6 Numerical examples 395

10.3 Internal Degrees of Preedom and Chemical Reactions 399

10.3.1 The generalized Borgnakke-Larsen model 399 10.3.2 Extensions to chemically

reacting flows 400 10.3.3 Numerical examples 402

10.4 Simulation Techniques on Parallel Computers 408

10.4.1 Simple parallel codes 408 10.4.2 Adaptive load balance techniques . . . . 409

10.5 References 415