Elsevier S & T

Handbook of Mathematical Fluid Dynamics

Friedlander, S.; Serre, D.

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Table of contents
  • Handbook of Mathematical Fluid Dynamicsiii
  • Copyrightiv
  • Contentsxi
  • Prefacev
  • List of Contributorsix
  • Chapter 1: The Boltzmann equation and fluid dynamics1
  • 1. Introduction3
  • 2. The basic molecular model4
  • 3. The Boltzmann equation5
  • 4. Molecules different from hard spheres9
  • 5. Collision invariants10
  • 6. The Boltzmann inequality and the Maxwell distributions12
  • 7. The macroscopic balance equations13
  • 8. The H-theorem17
  • 9. Model equations18
  • 10. The linearized collision operator21
  • 11. Boundary conditions22
  • 12. The continuum limit25
  • 13. Free-molecule and nearly free-molecule flows33
  • 14. Perturbations of equilibria36
  • 15. Approximate methods for linearized problems38
  • 16. Mixtures43
  • 17. Polyatomic gases46
  • 18. Chemistry and radiation52
  • 19. The DSMC method57
  • 20. Some applications of the DSMC method61
  • 21. Concluding remarks63
  • References63
  • Chapter 2: A review of mathematical topics in collisional kinetic theory71
  • Introduction73
  • Chapter 2A: General Presentation75
  • 1. Models for collisions in kinetic theory77
  • 2. Mathematical problems in collisional kinetic theory95
  • 3. Taxonomy118
  • 4. Basic surgery tools for the Boltzmann operator124
  • 5. Mathematical theories for the Cauchy problem130
  • Chapter 2B: Cauchy Problem141
  • 1. Use of velocity-averaging lemmas143
  • 2. Moment estimates147
  • 3. TheGrad’s cut-off toolbox153
  • 4. The singularity-hunter’s toolbox165
  • 5. The Landau approximation180
  • 6. Lower bounds185
  • Chapter 2C: H Theorem and Trend to Equilibrium189
  • 1. A gallery of entropy-dissipating kinetic models191
  • 2. Nonconstructive methods200
  • 3. Entropy dissipation methods203
  • 4. Entropy dissipation functionals of Boltzmann and Landau208
  • 5. Trend to equilibrium, spatially homogeneous Boltzmann and Landau224
  • 6. Gradient flows228
  • 7. Trend to equilibrium, spatially inhomogeneous systems235
  • Chapter 2D: Maxwell Collisions245
  • 1. Wild sums248
  • 2. Contracting probability metrics249
  • 3. Information theory254
  • 4. Conclusions258
  • Chapter 2E: OpenProblems and NewTrends263
  • 1. Open problems in classical collisional kinetic theory265
  • 2. Granularmedia272
  • 3. Quantum kinetic theory279
  • Bibliographical notes286
  • Acknowledgements287
  • References288
  • Chapter 3: Viscous and/or heat conducting compressible fluids307
  • 1. Basic equations of mathematical fluid dynamics309
  • 2. Mathematical aspects of the problem313
  • 3. The continuity equation and renormalized solutions321
  • 4. Weak convergence results325
  • 5. Mathematical theory of barotropic flows333
  • 6. Barotropic flows: large data existence results344
  • 7. Barotropic flows: asymptotic properties349
  • 8. Compressible–incompressible limits357
  • 9. Other topics, directions, alternative models361
  • 10. Conclusion364
  • References367
  • Chapter 4: Dynamic flows with liquid/vapor phase transitions373
  • 1. Introduction375
  • 2. The equations of motion378
  • 3. Initial value problems of the inviscid system (1.3) and admissibility criteria380
  • 4. Existence of solutions of theRiemann problem391
  • References417
  • Chapter 5: The Cauchy problem for the Euler equations for compressible fluids421
  • Abstract422
  • 1. Introduction423
  • 2. Local well-posedness for smooth solutions428
  • 3. Global well-posedness for smooth solutions432
  • 4. Formation of singularities in smooth solutions436
  • 5. Local well-posedness for discontinuous solutions443
  • 6. Global discontinuous solutions I: Riemann solutions446
  • 7. Global discontinuous solutions II: Glimm solutions458
  • 8. Global discontinuous solutions III: entropy solutions in BV479
  • 9. Global discontinuous solutions IV: entropy solutions in L∞488
  • 10. Global discontinuous solutions V: the multidimensional case513
  • 11. Euler equations for compressible fluids with source terms521
  • Acknowledgments531
  • References531
  • Chapter 6: Stability of strong discontinuities in fluids and MHD545
  • Abstract546
  • 1. Introduction547
  • 2. Basic steps of the stability analysis551
  • 3. Stability of gas dynamical shock waves574
  • 4. Stability of shock waves in relativistic gas dynamics597
  • 5. Stability of MHD shock waves606
  • 6. Stability of the MHD contact discontinuity629
  • 7. Rotational discontinuity in MHD634
  • 8. Instability of the MHD tangential discontinuity640
  • 9. Open problems646
  • References647
  • Chapter 7: On the motion of a rigid body in a viscous liquid: a mathematical analysis with applicati653
  • Introduction655
  • 1. Mathematical formulation665
  • 2. The liquid models669
  • Part I. Particle sedimentation671
  • 3. The free fall problem672
  • 4. Free fall in aNavier–Stokes liquid674
  • 5. Free fall in a second-order liquid737
  • Part II. Self-propelled bodies761
  • 6. The self-propelled body equations761
  • Acknowledgments787
  • References787
  • Author Index793
  • Subject Index807
Book details
  • Vendor Elsevier S & T
  • SKU 9780444503305
  • ISBN-13 9780080532929
  • Author Friedlander, S.; Serre, D.
  • Category Mathematics
  • Subject Applied

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The Handbook of Mathematical Fluid Dynamics is a compendium of essays that provides a survey of the major topics in the subject. Each article traces developments, surveys the results of the past decade, discusses the current state of knowledge and presents major future directions and open problems. Extensive bibliographic material is provided. The book is intended to be useful both to experts in the field and to mathematicians and other scientists who wish to learn about or begin research in mathematical fluid dynamics. The Handbook illuminates an exciting subject that involves rigorous mathematical theory applied to an important physical problem, namely the motion of fluids.