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Mathematical tools for kinetic equations


Author: Benoît Perthame
Translated by:
Journal: Bull. Amer. Math. Soc. 41 (2004), 205-244
MSC (2000): Primary 35F10, 35L60, 35Q35, 76P05, 82B40
DOI: https://doi.org/10.1090/S0273-0979-04-01004-3
Published electronically: February 2, 2004
MathSciNet review: 2043752
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Abstract: Since the nineteenth century, when Boltzmann formalized the concepts of kinetic equations, their range of application has been considerably extended. First introduced as a means to unify various perspectives on fluid mechanics, they are now used in plasma physics, semiconductor technology, astrophysics, biology.... They all are characterized by a density function that satisfies a Partial Differential Equation in the phase space.

This paper presents some of the simplest tools that have been devised to study more elaborate (coupled and nonlinear) problems. These tools are basic estimates for the linear first order kinetic-transport equation. Dispersive effects allow us to gain time decay, or space-time $L^p$ integrability, thanks to Strichartz-type inequalities. Moment lemmas gain better velocity integrability, and macroscopic controls transform them into space $L^p$integrability for velocity integrals.

These tools have been used to study several nonlinear problems. Among them we mention for example the Vlasov equations for mean field limits, the Boltzmann equation for collisional dilute flows, and the scattering equation with applications to cell motion (chemotaxis).

One of the early successes of kinetic theory has been to recover macroscopic equations from microscopic descriptions and thus to be able theoretically to compute transport coefficients. We also present several examples of the hydrodynamic limits, the diffusion limits and especially the recent derivation of the Navier-Stokes system from the Boltzmann equation, and the theory of strong field limits.


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  • 1. Agoshkov, V. I. Spaces of functions with differential-difference characteristics and smoothness of solutions of the transport equation. Soviet Math. Dokl. 29 (1984), 662-666. MR 86a:46033
  • 2. Alexandre, R.; Desvillettes, L.; Villani, C.; Wennberg, B. Entropy dissipation and long-range interactions. Arch. Ration. Mech. Anal. 152 (2000), no. 4, 327-355. MR 2001c:82061
  • 3. Allaire, G.; Bal, G. Homogenization of the criticality spectral equation in neutron transport. M2AN Math. Model. Numer. Anal. 33 (1999), no. 4, 721-746.MR 2001c:35025
  • 4. Arkeryd, L.; Cercignani, C. Global existence in $L^1$ for the Enskog equation and convergence of the solutions to solutions of the Boltzmann equation. J. Statist. Phys. 59 (1990), no. 3-4, 845-867. MR 91m:82114
  • 5. Arkeryd, L.; Nouri, A. $L^1$ solutions to the stationary Boltzmann equation in a slab. Ann. Fac. Sci. Toulouse Math. (6) 9 (2000), no. 3, 375-413. MR 2002d:82061
  • 6. Arnold, A.; Carrillo, J. A.; Gamba, I. M.; Shu, C.-W. Low and high field scaling limits for the Vlasov- and Wigner-Poisson-Fokker-Planck systems. The Sixteenth International Conference on Transport Theory, Part I (Atlanta, GA, 1999). Transport Theory Statist. Phys. 30 (2001), no. 2-3, 121-153. MR 2002h:82080
  • 7. Babovsky, H.; Bardos, C.; P\latkowski, T. Diffusion approximation for a Knudsen gas in a thin domain with accommodation on the boundary. Asymptotic Anal. 3 (1991), no. 4, 265-289. MR 91m:76095
  • 8. Bardos, C.; Degond, P. Global existence for the Vlasov-Poisson equation in $3$ space variables with small initial data. Ann. Inst. H. Poincaré Anal. Non Linéaire 2 (1985), no. 2, 101-118. MR 86k:35129
  • 9. Bardos, C.; Golse, F.; Levermore, C. D. Fluid dynamic limits of kinetic equations. II. Convergence proofs for the Boltzmann equation. Comm. Pure Appl. Math. 46 (1993), no. 5, 667-753. MR 94g:82039
  • 10. Bardos, C.; Golse, F.; Perthame, B.; Sentis, R. The nonaccretive radiative transfer equations: existence of solutions and Rosseland approximation. J. Funct. Anal. 77 (1988), no. 2, 434-460. MR 89f:35174
  • 11. Bardos, C.; Santos, R.; Sentis, R. Diffusion approximation and computation of the critical size. Trans. Amer. Math. Soc. 284 (1984), no. 2, 617-649. MR 86g:45027
  • 12. Batt, J.; Rein, G. Global classical solutions of the periodic Vlasov-Poisson system in three dimensions. C. R. Acad. Sci. Paris Sér. I Math. 313 (1991), no. 6, 411-416. MR 92f:35138
  • 13. Batt, J.; Rein, G. A rigorous stability result for the Vlasov-Poisson system in three dimensions. Ann. Mat. Pura Appl. (4) 164 (1993), 133-154. MR 95a:82106
  • 14. Bellomo, N; Preziosi, L. Modeling and mathematical problems related to tumors immune system interactions. Math. Comp. Modelling 32 (2000), 413-452. MR 2001i:92016
  • 15. Bellomo, N; Pulvirenti, M. Modeling in applied sciences. A kinetic theory approach. Modeling and Simulation in Science, Engineering and Technology. Birkhauser, Boston (2000). MR 2001a:82003
  • 16. Ben Abdallah, N.; Degond, P. The Child-Langmuir law in the kinetic theory of charged particles: semiconductors models. Mathematical problems in semiconductor physics (Rome, 1993), 76-102, Pitman Res. Notes Math. Ser., 340, Longman, Harlow, 1995. MR 98j:82074
  • 17. Benamou, J.D.; Castella, F.; Katsaounis, T.; Perthame, B. High frequency limit of the Helmholtz equations, Rev. Mat. IberoAmer. 18 (2002). MR 2003h:35262
  • 18. Benedetto, D.; Caglioti, E.; Golse, F.; Pulvirenti, M. A hydrodynamic model arising in the context of granular media. Comput. Math. Appl. 38 (1999), no. 7-8, 121-131. MR 2000g:76092
  • 19. Benedetto, D.; Pulvirenti, M. On the one-dimensional Boltzmann equation for granular flows. M2AN Math. Model. Numer. Anal. 35 (2001), no. 5, 899-905. MR 2002h:82066
  • 20. Bensoussan, A.; Lions, J.-L.; Papanicolaou, G. Boundary layers and homogenization of transport processes, RIMS, Kyoto Univ. 15 (1979), 53-157. MR 80i:60122
  • 21. Bézard, M. Régularité précisée des moyennes dans les équations de transport. Bull. Soc. Math. France 122 (1994), 29-76. MR 95g:82083
  • 22. Bobylev, A. V.; Carrillo, J. A.; Gamba, I. M. On some properties of kinetic and hydrodynamic equations for inelastic interactions. J. Statist. Phys. 98 (2000), no. 3-4, 743-773. MR 2001c:82063
  • 23. Bouchut, F.; Smoothing effect for the non-linear Vlasov-Poisson-Fokker-Planck system. J. Differential Equations 122 (1995), no. 2, 225-238. MR 96g:35198
  • 24. Bouchut, F.; Desvillettes, L. Averaging lemmas without time Fourier transform and application to discretized kinetic equations. Proc. Roy. Soc. Edinburgh Sect. A 129 (1999), no. 1, 19-36. MR 2000b:82024
  • 25. Bouchut, F.; Dolbeault, J. On long time asymptotics of the Vlasov-Fokker-Planck equation and of the Vlasov-Poisson-Fokker-Planck system with Coulombic and Newtonian potentials. Differential Integral Equations 8 (1995), no. 3, 487-514. MR 95i:82076
  • 26. Bouchut, F.; Golse, F.; Pulvirenti, M. in Kinetic Equations and Asymptotic Theories, L. Desvillettes and B. Perthame ed., Series in Appl. Math. no. 4, Elsevier (2000).
  • 27. Bournaveas, N.; Perthame, B. Averages over spheres for kinetic transport equations; hyperbolic Sobolev spaces and Strichartz inequalities. J. Math. Pures Appl. (9) 80 (2001), no. 5, 517-534. MR 2002b:82051
  • 28. Brenier, Y. Convergence of the Vlasov-Poisson system to the incompressible Euler equations. Comm. Partial Differential Equations 25 (2000), no. 3-4, 737-754. MR 2001c:76124
  • 29. Caflisch, R. E. The fluid dynamic limit of the nonlinear Boltzmann equation. Comm. Pure Appl. Math. 33 (1980), no. 5, 651-666. MR 81j:76072
  • 30. Caprino, S.; Marchioro, C.; Pulvirenti, M. On the two-dimensional Vlasov-Helmholtz equation with infinite mass. Comm. Partial Differential Equations 27 (2002), no. 3-4, 791-808. MR 2003b:82056
  • 31. Castella, F.; Perthame, B. Strichartz estimates for kinetic transport equations. C. R. Acad. Sc. Série I Math. 322, no. 6 (1996), 535-540. MR 98f:35133
  • 32. Cercignani, C.; The Boltzmann equation and its application. Applied Math. Sciences 67, Springer-Verlag, Berlin (1988). MR 95i:82082
  • 33. Cercignani, C.; Illner, R.; Pulvirenti, M. The mathematical theory of dilute gases. Applied Math. Sciences 106, Springer-Verlag, Berlin (1994). MR 96g:82046
  • 34. Chalub, F.; Markowich, M.; Perthame, B.; Schmeiser, C. Kinetic Models for Chemotaxis and their Drift-Diffusion Limits. Preprint (2002).
  • 35. Chen, G.-Q.; Perthame, B. Well-posedness for non-isotropic degenerate parabolic-hyperbolic equations. Ann. Inst. H. Poincaré Anal. Non Lineaire 20 (2003), no. 4, 645-668.
  • 36. Dautray, R.; Lions, J.-L. Mathematical analysis and numerical methods for science and technology. Springer-Verlag, Berlin, 1992. MR 92k:00006
  • 37. Decoster, A.; Markowich, P. A.; Perthame, B. Modeling of collisions. With contributions by I. Gasser, A. Unterreiter and L. Desvillettes. Edited and with a foreword by P. A. Raviart. Series in Applied Mathematics, 2. Gauthier-Villars, Éditions Scientifiques et Médicales Elsevier, Paris; North-Holland, Amsterdam, 1998. MR 2000b:76092
  • 38. Degond, P. Mathematical modelling of microelectronics semiconductor devices. Some current topics on nonlinear conservation laws, 77-110, AMS/IP Stud. Adv. Math., 15, Amer. Math. Soc., Providence, RI (2000). MR 2001h:82108
  • 39. Degond, P.; Goudon, T.; Poupaud, F. Diffusion limit for nonhomogeneous and non-micro-reversible processes. Indiana Univ. Math. J. 49 (2000), no. 3, 1175-1198. MR 2002a:35012
  • 40. Degond, P.; Jüngel, A. High-field approximations of the energy-transport model for semiconductors with non-parabolic band structure. Z. Angew. Math. Phys. 52 (2001), no. 6, 1053-1070. MR 2002j:82120
  • 41. Degond, P.; Lucquin-Desreux, B. The Fokker-Plansk asymptotics of the Boltzmann collision operator in the Coulomb case. Math. Mod. Meth. Appl. Sc. 2 (1992), no. 2, 167-182. MR 93g:82083
  • 42. Desvillettes, L.; Dolbeault, J. On long time asymptotics of the Vlasov-Poisson-Boltzmann equation. Comm. Partial Differential Equations 16 (1991), no. 2-3, 451-489. MR 92b:35153
  • 43. Desvillettes, L.; Villani, C. On the trend to global equilibrium in spatially inhomogeneous entropy-dissipating systems: the linear Fokker-Planck equation. Comm. Pure Appl. Math. 54 (2001), no. 1, 1-42. MR 2001h:82079
  • 44. Desvillettes, L.; Villani, C. On the trend to equilibrium in spatially inhomogeneous entropy-dissipating systems: the Boltzmann equation. Preprint (2002). MR 2003g:82085
  • 45. DeVore R.; Petrova G. P. The averaging lemma, J. Amer. Math. Soc. 14 (2001), no. 2, 279-296 (electronic). MR 2002b:35130
  • 46. DiPerna, R.; Lions, P.-L. Solutions globales d'équations du type Vlasov-Poisson. (French) [Global solutions of Vlasov-Poisson type equations] C. R. Acad. Sci. Paris Sér. I Math. 307 (1988), no. 12, 655-658. MR 89k:35202
  • 47. DiPerna, R. J.; Lions, P.-L. Global weak solutions of Vlasov-Maxwell systems. Comm. Pure Appl. Math. 42 (1989), no. 6, 729-757. MR 90i:35236
  • 48. DiPerna, R. J.; Lions, P.-L. Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 98 (1989), no. 3, 511-547. MR 90j:34004
  • 49. DiPerna, R. J.; Lions, P.-L. On the Cauchy problem for the Boltzmann equation: global existence and weak stability results. Annals of Math 130 (1989), 321-366. MR 90k:82045
  • 50. DiPerna, R. J.; Lions, P.-L.; Meyer, Y. $L^p$ regularity of velocity averages. Ann. Inst. H. Poincaré, Analyse non-linéaire 8 (1991), no.3-4, 271-287. MR 92g:35036
  • 51. Dolbeault, J. Free energy and solutions of the Vlasov-Poisson-Fokker-Planck system: external potential and confinement (large time behavior and steady states). J. Math. Pures Appl. (9) 78 (1999), no. 2, 121-157. MR 99m:35248
  • 52. Dubroca, B.; Feugeas, J.-L. Etude théorique et numérique d'une hiérarchie de modèles aux moments pour le transfert radiatif. (French) [Theoretical and numerical study of a moment closure hierarchy for the radiative transfer equation] C. R. Acad. Sci. Paris Sér. I Math. 329 (1999), no. 10, 915-920. MR 2000h:85005
  • 53. Escobedo, M.; Mischler, S. On a quantum Boltzmann equation for a gas of photons. J. Math. Pures Appl. (9) 80 (2001), no. 5, 471-515. MR 2002j:82071
  • 54. Evans, L. C. Partial Differential Equations, Graduate Studies in Mathematics Vol. 19, American Mathematical Society (1998). MR 99e:35001
  • 55. Frénod, E.; Sonnendrücker, E. The finite Larmor radius approximation. SIAM J. Math. Anal. 32 (2001), no. 6, 1227-1247 (electronic). MR 2002g:82049
  • 56. Gasser, I.; Jabin, P.-E.; Perthame, B. Regularity and propagation of moments in some nonlinear Vlasov systems. Proc. Roy. Soc. Edinburgh Sect. A 130 (2000), no. 6, 1259-1273. MR 2001j:82101
  • 57. Gérard, P. Moyennisation et régularité deux-microlocale. (French) [Second-microlocal averaging and regularity]. Ann. Sci. École Norm. Sup. (4) 23 (1990), no. 1, 89-121. MR 91g:35068
  • 58. Gérard, P.; Golse, F. Averaging regularity results for PDEs under transversality assumptions. Comm. Pure Appl. Math. 45 (1992), no. 1, 1-26. MR 92k:35044
  • 59. Gérard, P.; Markowich, P. A.; Mauser, N. J.; Poupaud, F. Homogenization limits and Wigner transforms. Comm. Pure Appl. Math. 50 (1997), no. 4, 323-379. MR 98d:35020
  • 60. Ginibre, J.; Velo, G. The global Cauchy problem for some nonlinear Schrödinger equation revisited, Ann. Inst. H. Poincaré, Analyse non linéaire 2 (1985) 309-327. MR 87b:35150
  • 61. Glassey, R. T. The Cauchy problem in kinetic theory, SIAM, Philadelphia (1996). MR 97i:82070
  • 62. Glassey, R. T.; Schaeffer, J. The relativistic Vlasov-Maxwell system in 2D and 2.5D. Nonlinear wave equations (Providence, RI, 1998), 61-69, Contemp. Math., 263, Amer. Math. Soc., Providence, RI, 2000. MR 2001f:35247
  • 63. Glassey, R. T.; Strauss, W. A. Asymptotic stability of the relativistic Maxwellian via fourteen moments. Transport Theory Statist. Phys. 24 (1995), no. 4-5, 657-678. MR 96c:82054
  • 64. Glassey, R. T.; Strauss, W. A. Decay of the linearized Boltzmann-Vlasov system. Transport Theory Statist. Phys. 28 (1999), no. 2, 135-156. MR 2000c:35238
  • 65. Golse, F.; Lions, P.-L.; Perthame, B.; Sentis, R. Regularity of the moments of the solution of a transport equation. J. Funct. Anal. 76 (1988) 110-125. MR 89a:35179
  • 66. Golse, F.; Perthame, B.; Sentis, R. Un résultat de compacité pour les équations du transport$\ldots$, C. R. Acad. Sci. Paris, Série I Math. 301 (1985) 341-344. MR 86m:82062
  • 67. Golse, F.; Saint-Raymond, L. The Vlasov-Poisson system with strong magnetic field. J. Math. Pures Appl. (9) 78 (1999), no. 8, 791-817. MR 2000g:35209
  • 68. Golse, F.; Saint-Raymond, L. The Navier-Stokes limit of Boltzmann equation: convergence proof. Inventiones Mathematicae 155, no. 1 (2004), 81-161.
  • 69. Goudon, T.; Jabin, P.-E.; Vasseur, A. Hydrodynamic limits for the Vlasov-Navier-Stokes equations: hyperbolic scaling, preprint Ecole Normale Supérieure, DMA 02-29 and parabolic scaling 02-30. http://www.dma.ens.fr/edition/publis/2002.
  • 70. Goudon, T.; Poupaud, F. Approximation by homogenization and diffusion of kinetic equations. Comm. Partial Differential Equations 26 (2001), no. 3-4, 537-569. MR 2002e:35195
  • 71. Grenier, E. Oscillations in quasineutral plasmas. Comm. Partial Differential Equations 21 (1996), no. 3-4, 363-394. MR 98g:82033
  • 72. Guo, Y.; Strauss, W. A. Unstable BGK solitary waves and collisionless shocks. Comm. Math. Phys. 195 (1998), no. 2, 267-293. MR 2000k:35239b
  • 73. Guo, Y.; Strauss, W. A. Magnetically created instability in a collisionless plasma. J. Math. Pures Appl. (9) 79 (2000), no. 10, 975-1009. MR 2001k:76053
  • 74. Hamdache, K. Global existence and large time behaviour of solutions for Vlasov-Stokes equations. Japan J. Indust. Appl. Math. 15 (1998), 51-74. MR 99f:76128
  • 75. Horst, E. On the classical solutions of the initial value problem for the unmodified non-linear Vlasov equation I. Math. Meth. Appl. Sci. 3 (1981), 229-248; and II. Math. Meth. Appl. Sci. 4 (1982), 19-32. MR 83h:35110, MR 84b:85001
  • 76. Horst, E.; Hunze, R. Weak solutions of the initial value problem for the unmodified nonlinear Vlasov equation. Math. Methods Appl. Sci. 6 (1984), no. 2, 262-279. MR 85j:45026
  • 77. Illner, R.; Klar, A.; Lange, H.; Unterreiter, A.; Wegener, R. A kinetic model for vehicular traffic: existence of stationary solutions. J. Math. Anal. Appl. 237 (1999) no. 2, 622-643. MR 2000e:90010
  • 78. Illner, R.; Rein, G. Time decay of the solutions of the Vlasov-Poisson system in the plasma physical case. Math. Methods Appl. Sci. 19 (1996), no. 17, 1409-1413. MR 97j:35153
  • 79. Jabin, P.-E. Macroscopic limit of Vlasov type equations with friction. Ann. Inst. H. Poincaré Anal. Non Linéaire 17 (2000), no. 5, 651-672. MR 2002i:82079
  • 80. Jabin, P.-E.; Perthame, B. Notes on mathematical problems on the dynamics of dispersed particles interacting through a fluid. In N. Bellomo and M. Pulvirenti editors, Modelling in applied sciences, a kinetic theory approach, Birkhäuser, Boston (2000). MR 2002b:76098
  • 81. Jabin, P.-E.; Perthame, B. Regularity in kinetic formulations via averaging lemmas. ESAIM:COCV special issue in memory of J.-L. Lions (2002), ESAIM Control Optim. Calc. Var. 8 (2002), 761-774 (electronic). MR 2003j:35207
  • 82. Jabin, P.-E.; Vega, L. A real space method for averaging lemmas. Preprint ENS-DMA 03-09 (2003).
  • 83. Junk, M. Domain of definition of Levermore's five-moment system. J. Stat. Phys. 93 (1998), no. 5-6, 1143-1167. MR 99k:82063
  • 84. Kawashima, S.; Matsumura, A.; Nishida, T. On the fluid-dynamical approximation to the Boltzmann equation at the level of the Navier-Stokes equation. Comm. Math. Phys. 70 (1979), no. 2, 97-124. MR 81b:76048
  • 85. Keel, M.; Tao, T. Endpoint Strichartz estimates. Amer. J. Math. 120 (1998), no. 5, 955-980. MR 2000d:35018
  • 86. Levermore, C. D. Moment closure hierarchies for kinetic theories. J. Statist. Phys. 83 (1996), no. 5-6, 1021-1065. MR 97e:82041
  • 87. Lions, P.-L.; Compactness in Boltzmann Equation via Fourier integral operators and applications. Part I and II J. Math. Kyoto Univ., 34(2) (1994) 1-61 and Part III J. Math. Kyoto Univ., 34(3) (1994) 539-584. MR 96f:35133, MR 96f:35134
  • 88. Lions, P.-L. Régularité optimale des moyennes en vitesses. II. (French) [Optimal regularity of velocity averages. II] C. R. Acad. Sci. Paris Sér. I Math. 326 (1998), no. 8, 945-948. MR 99m:35252
  • 89. Lions, P.-L.; Masmoudi, N. From the Boltzmann equations to the equations of incompressible fluid mechanics. I, II. Arch. Ration. Mech. Anal. 158 (2001), no. 3, 173-193, 195-211 MR 2002m:76085
  • 90. Lions, P.-L.; Perthame, B. Propagation of moments and regularity for the $3$-dimensional Vlasov-Poisson system. Invent. Math. 105 (1991), no. 2, 415-430. MR 92e:35160
  • 91. Lions , P.-L.; Perthame, B. Lemmes de moments, de moyenne et de dispersion, C. R. Acad. Sci. Paris, Série I Math. 314 (1992), 801-806. MR 93f:35217
  • 92. Lions , P.-L.; Perthame, B.; Tadmor, E. A kinetic formulation of multidimensional scalar conservation laws and related equations. J. Amer. Math. Soc. 7 (1994), 169-191. MR 94d:35100
  • 93. Liu, T.-P.; Yu, S.-H. Boltzmann equation: micro-macro decompositions and positivity or shock profiles. Preprint 2002.
  • 94. Markowich, P. A.; Ringhofer, C. A.; Schmeiser, C. Semiconductor equations. Springer-Verlag, Vienna, 1990. MR 91j:78011
  • 95. Mischler, S. On the initial boundary value problem for the Vlasov-Poisson-Boltzmann system. Comm. Math. Phys. 210 (2000), no. 2, 447-466. MR 2001f:45013
  • 96. Mischler, S. On weak-weak convergences and some applications to the initial boundary value problem for kinetic equations. Preprint (2001).
  • 97. Mischler, S.; Wennberg, B. On the spatially homogeneous Boltzmann equation. Ann. Inst. H. Poincaré Anal. Non Linéaire 16 (1999), no. 4, 467-501. MR 2000f:35147
  • 98. Nieto, J.; Poupaud, F.; Soler, J. High field limit for the Vlasov-Poisson-Fokker-Planck system. Arch. Ration. Mech. Anal. 158 (2001), no. 1, 29-59. MR 2002g:82050
  • 99. Othmer, H. G.; Dunbar, S. R.; Alt, W. Models of dispersal in biological systems, J. Math. Biol., 26(3) 263-298 (1988). MR 90a:92064
  • 100. Papanicolaou, G.; Ryzhik, L. Waves and Transport. IAS/ Park City Mathematics series. Volume 5, Amer. Math. Soc., Providence, RI, 1999. MR 99k:73063
  • 101. Perthame, B. Time decay, propagation of low moments and dispersive effects for kinetic equations. Comm. Partial Differential Equations 21 (1996), no. 3-4, 659-686. MR 97e:82042
  • 102. Perthame, B. Kinetic formulation of conservation laws. Oxford Univ. Press, Series in Math. and Appl. 21 (2002).
  • 103. Perthame, B.; Souganidis, P. E. A limiting case of velocity averaging lemmas. Ann. Sc. E.N.S. Série 4, 31 (1998) 591-598. MR 99h:82064
  • 104. Perthame, B.; Vega, L. Morrey-Campanato estimates for Helmholtz Equations. J. Funct. Anal. 164 (2) (1999), 340-355. MR 2000i:35023
  • 105. Perthame, B.; Vega, L. Sommerfeld condition for a Liouville equation and concentration of trajectories. Bull. of Braz. Math. Soc., New Series 34(1), 1-15 (2003).
  • 106. Poupaud, F. Runaway phenomena and fluid approximation under high fields in semiconductor kinetic theory. Z. Angew. Math. Mech. 72 (1992), no. 8, 359-372. MR 93h:82074
  • 107. Pfaffelmoser, K. Global classical solutions of the Vlasov-Poisson system in three dimensions for general initial data. J. Differential Equations 95 (1992), no. 2, 281-303. MR 93d:35170
  • 108. Russo, G.; Smereka, P. Kinetic theory for bubbly flow I: collisionless case and II: fluid dynamic limit. SIAM J. Appl. Math., 56 (1996), no.2, 327-357 and 358-371. MR 97f:82042, MR 97f:82043
  • 109. Saint-Raymond, L. The gyrokinetic approximation for the Vlasov-Poisson system. Math. Models Methods Appl. Sci. 10 (2000), no. 9, 1305-1332. MR 2001h:82096
  • 110. Schaeffer, J. Global existence of smooth solutions to the Vlasov-Poisoon system in three dimensions, Comm. P.D.E., 16, 1313-1335 (1991). MR 92g:82113
  • 111. Sone, Y. Kinetic theory and fluid dynamics. Birkhäuser, Boston (2002). MR 2003h:76113
  • 112. Toscani, G. One-dimensional kinetic models of granular flows. M2AN Math. Model. Numer. Anal. 34 (2000), no. 6, 1277-1291. MR 2002a:82094
  • 113. Truesdell, C.; Muncaster, R. G. Fundamentals of Maxwell's kinetic theory of a simple monatomic gas. Treated as a branch of rational mechanics. Pure and Applied Mathematics, 83. Academic Press, New York-London, 1980. MR 81f:80001
  • 114. Ukai, S. The Boltzmann-Grad limit and Cauchy-Kovalevskaya theorem. Recent topics in mathematics moving toward science and engineering. Japan J. Indust. Appl. Math. 18 (2001), no. 2, 383-392. MR 2002e:82057
  • 115. Ukai, S.; Asano, K. Steady solutions of the Boltzmann equation for a gas flow past an obstacle. I. Existence. Arch. Rational Mech. Anal. 84 (1983), no. 3, 249-291. MR 85a:76079
  • 116. Villani, C. A review of mathematical topics in collisional kinetic theory. Handbook of mathematical fluid dynamics, Vol. I, 71-305, North-Holland, Amsterdam, 2002. MR 2003k:82087
  • 117. Villani, C. Cercignani's conjecture is sometimes true and always almost true. Comm. Math. Phys. 234 (2003), no. 3, 455-490.
  • 118. Westdickenberg, M. Some new velocity averaging results. SIAM J. Math. Anal. 33 (2002), no. 5, 1007-1032 (electronic). MR 2003b:35027

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Additional Information

Benoît Perthame
Affiliation: Département de Mathématiques et Applications, CNRS UMR 8553, École Normale Supérieure, 45, rue d’Ulm, 75230 Paris Cedex 05, France
Email: Benoit.Perthame@ens.fr

DOI: https://doi.org/10.1090/S0273-0979-04-01004-3
Received by editor(s): January 15, 2003
Received by editor(s) in revised form: November 6, 2003
Published electronically: February 2, 2004
Article copyright: © Copyright 2004 American Mathematical Society

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