# zbMATH — the first resource for mathematics

Mathematical tools for kinetic equations. (English) Zbl 1151.82351
The author presents general methods for linear kinetic equations, which covers time decay and dispersion effects as Strichartz inequalities moment lemmas – all of these improve the obvious integrability derived from conservation laws. It is well-known that in the context of the linear equation it is possible to gain regularity to averaging lemmas. The author gives several statements beginning with the simplest regularizing effect $$H^{1/2}$$. These tools have been used to treat nonlinear models in the last few years. The author presents in this context the Vlasov equation of plasma physics, scattering models and the Boltzmann equation. Moreover, the author deals with also asymptotic problems and the derivation of macroscopic models especially through the diffusion, hyperbolic and high field limits.

##### MSC:
 82B40 Kinetic theory of gases in equilibrium statistical mechanics 76P05 Rarefied gas flows, Boltzmann equation in fluid mechanics 35F10 Initial value problems for linear first-order PDEs
Full Text:
##### References:
 [1] V. I. Agoshkov, Spaces of functions with differential-difference characteristics and the smoothness of solutions of the transport equation, Dokl. Akad. Nauk SSSR 276 (1984), no. 6, 1289 – 1293 (Russian). [2] R. Alexandre, L. Desvillettes, C. Villani, and B. Wennberg, Entropy dissipation and long-range interactions, Arch. Ration. Mech. Anal. 152 (2000), no. 4, 327 – 355. · Zbl 0968.76076 [3] Grégoire Allaire and Guillaume Bal, Homogenization of the criticality spectral equation in neutron transport, M2AN Math. Model. Numer. Anal. 33 (1999), no. 4, 721 – 746 (English, with English and French summaries). · Zbl 0931.35010 [4] Leif Arkeryd and Carlo Cercignani, Global existence in \?\textonesuperior 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. · Zbl 0780.76066 [5] Leif Arkeryd and Anne Nouri, \?\textonesuperior solutions to the stationary Boltzmann equation in a slab, Ann. Fac. Sci. Toulouse Math. (6) 9 (2000), no. 3, 375 – 413 (English, with English and French summaries). · Zbl 0991.45005 [6] A. Arnold, J. A. Carrillo, I. Gamba, and C.-W. Shu, Low and high field scaling limits for the Vlasov- and Wigner-Poisson-Fokker-Planck systems, Transport Theory Statist. Phys. 30 (2001), no. 2-3, 121 – 153. The Sixteenth International Conference on Transport Theory, Part I (Atlanta, GA, 1999). · Zbl 1106.82381 [7] Hans Babovsky, Claude Bardos, and Tadeusz Płatkowski, Diffusion approximation for a Knudsen gas in a thin domain with accommodation on the boundary, Asymptotic Anal. 3 (1991), no. 4, 265 – 289. · Zbl 0850.76599 [8] C. Bardos and P. Degond, 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 (English, with French summary). · Zbl 0593.35076 [9] Claude Bardos, François Golse, and C. David Levermore, Fluid dynamic limits of kinetic equations. II. Convergence proofs for the Boltzmann equation, Comm. Pure Appl. Math. 46 (1993), no. 5, 667 – 753. · Zbl 0817.76002 [10] C. Bardos, F. Golse, B. Perthame, and R. Sentis, The nonaccretive radiative transfer equations: existence of solutions and Rosseland approximation, J. Funct. Anal. 77 (1988), no. 2, 434 – 460. · Zbl 0655.35075 [11] C. Bardos, R. Santos, and R. Sentis, Diffusion approximation and computation of the critical size, Trans. Amer. Math. Soc. 284 (1984), no. 2, 617 – 649. · Zbl 0508.60067 [12] Jürgen Batt and Gerhard Rein, 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 (English, with French summary). · Zbl 0741.35058 [13] Jürgen Batt and Gerhard Rein, A rigorous stability result for the Vlasov-Poisson system in three dimensions, Ann. Mat. Pura Appl. (4) 164 (1993), 133 – 154. · Zbl 0791.49030 [14] N. Bellomo and L. Preziosi, Modelling and mathematical problems related to tumor evolution and its interaction with the immune system, Math. Comput. Modelling 32 (2000), no. 3-4, 413 – 452. · Zbl 0997.92020 [15] Nicola Bellomo and Mario Pulvirenti , Modeling in applied sciences, Modeling and Simulation in Science, Engineering and Technology, Birkhäuser Boston, Inc., Boston, MA, 2000. A kinetic theory approach. · Zbl 0957.76075 [16] Naoufel Ben Abdallah and Pierre Degond, The Child-Langmuir law in the kinetic theory of charged particles: semiconductors models, Mathematical problems in semiconductor physics (Rome, 1993) Pitman Res. Notes Math. Ser., vol. 340, Longman, Harlow, 1995, pp. 76 – 102. · Zbl 0888.35114 [17] Jean-David Benamou, François Castella, Theodoros Katsaounis, and Benoit Perthame, High frequency limit of the Helmholtz equations, Rev. Mat. Iberoamericana 18 (2002), no. 1, 187 – 209. · Zbl 1090.35165 [18] D. Benedetto, E. Caglioti, F. Golse, and M. Pulvirenti, A hydrodynamic model arising in the context of granular media, Comput. Math. Appl. 38 (1999), no. 7-8, 121 – 131. · Zbl 0946.76096 [19] Dario Benedetto and Mario Pulvirenti, On the one-dimensional Boltzmann equation for granular flows, M2AN Math. Model. Numer. Anal. 35 (2001), no. 5, 899 – 905. · Zbl 1044.82012 [20] Alain Bensoussan, Jacques-L. Lions, and George C. Papanicolaou, Boundary layers and homogenization of transport processes, Publ. Res. Inst. Math. Sci. 15 (1979), no. 1, 53 – 157. · Zbl 0408.60100 [21] Max Bézard, Régularité \?^{\?} précisée des moyennes dans les équations de transport, Bull. Soc. Math. France 122 (1994), no. 1, 29 – 76 (French, with English and French summaries). · Zbl 0798.35025 [22] A. V. Bobylev, J. A. Carrillo, and I. M. Gamba, On some properties of kinetic and hydrodynamic equations for inelastic interactions, J. Statist. Phys. 98 (2000), no. 3-4, 743 – 773. · Zbl 1056.76071 [23] François Bouchut, Smoothing effect for the non-linear Vlasov-Poisson-Fokker-Planck system, J. Differential Equations 122 (1995), no. 2, 225 – 238. · Zbl 0840.35053 [24] F. Bouchut and L. Desvillettes, Averaging lemmas without time Fourier transform and application to discretized kinetic equations, Proc. Roy. Soc. Edinburgh Sect. A 129 (1999), no. 1, 19 – 36. · Zbl 0933.35159 [25] F. Bouchut and J. Dolbeault, 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. · Zbl 0830.35129 [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] Nikolaos Bournaveas and Benoit Perthame, Averages over spheres for kinetic transport equations; hyperbolic Sobolev spaces and Strichartz inequalities, J. Math. Pures Appl. (9) 80 (2001), no. 5, 517 – 534. · Zbl 1036.82023 [28] Y. Brenier, Convergence of the Vlasov-Poisson system to the incompressible Euler equations, Comm. Partial Differential Equations 25 (2000), no. 3-4, 737 – 754. · Zbl 0970.35110 [29] Russel E. Caflisch, The fluid dynamic limit of the nonlinear Boltzmann equation, Comm. Pure Appl. Math. 33 (1980), no. 5, 651 – 666. · Zbl 0424.76060 [30] S. Caprino, C. Marchioro, and M. Pulvirenti, On the two-dimensional Vlasov-Helmholtz equation with infinite mass, Comm. Partial Differential Equations 27 (2002), no. 3-4, 791 – 808. · Zbl 1007.35094 [31] François Castella and Benoît Perthame, Estimations de Strichartz pour les équations de transport cinétique, C. R. Acad. Sci. Paris Sér. I Math. 322 (1996), no. 6, 535 – 540 (French, with English and French summaries). · Zbl 0848.35095 [32] Carlo Cercignani, The Boltzmann equation and its applications, Applied Mathematical Sciences, vol. 67, Springer-Verlag, New York, 1988. · Zbl 0646.76001 [33] Carlo Cercignani, Reinhard Illner, and Mario Pulvirenti, The mathematical theory of dilute gases, Applied Mathematical Sciences, vol. 106, Springer-Verlag, New York, 1994. · Zbl 0813.76001 [34] Chalub, F.; Markowich, M.; Perthame, B.; Schmeiser, C. Kinetic Models for Chemotaxis and their Drift-Diffusion Limits. Preprint (2002). · Zbl 1052.92005 [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. · Zbl 1031.35077 [36] Robert Dautray and Jacques-Louis Lions, Mathematical analysis and numerical methods for science and technology. Vol. 5, Springer-Verlag, Berlin, 1992. Evolution problems. I; With the collaboration of Michel Artola, Michel Cessenat and Hélène Lanchon; Translated from the French by Alan Craig. · Zbl 0755.35001 [37] A. Decoster, P. A. Markowich, and B. Perthame, Modeling of collisions, Series in Applied Mathematics (Paris), vol. 2, Gauthier-Villars, Éditions Scientifiques et Médicales Elsevier, Paris; North-Holland, Amsterdam, 1998. With contributions by I. Gasser, A. Unterreiter and L. Desvillettes; Edited and with a foreword by P. A. Raviart. · Zbl 0924.76002 [38] Pierre Degond, Mathematical modelling of microelectronics semiconductor devices, Some current topics on nonlinear conservation laws, AMS/IP Stud. Adv. Math., vol. 15, Amer. Math. Soc., Providence, RI, 2000, pp. 77 – 110. · Zbl 0956.35121 [39] P. Degond, T. Goudon, and F. Poupaud, Diffusion limit for nonhomogeneous and non-micro-reversible processes, Indiana Univ. Math. J. 49 (2000), no. 3, 1175 – 1198. · Zbl 0971.82035 [40] Pierre Degond and Ansgar Jüngel, 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. · Zbl 0991.35043 [41] P. Degond and B. Lucquin-Desreux, The Fokker-Planck asymptotics of the Boltzmann collision operator in the Coulomb case, Math. Models Methods Appl. Sci. 2 (1992), no. 2, 167 – 182. · Zbl 0755.35091 [42] L. Desvillettes and J. Dolbeault, On long time asymptotics of the Vlasov-Poisson-Boltzmann equation, Comm. Partial Differential Equations 16 (1991), no. 2-3, 451 – 489. · Zbl 0737.35127 [43] L. Desvillettes and C. Villani, 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. , https://doi.org/10.1002/1097-0312(200101)54:13.0.CO;2-Q · Zbl 1029.82032 [44] C. Villani, On the trend to equilibrium for kinetic equations, Markov Process. Related Fields 8 (2002), no. 2, 237 – 250. Inhomogeneous random systems (Cergy-Pontoise, 2001). · Zbl 1128.82303 [45] Ronald DeVore and Guergana Petrova, The averaging lemma, J. Amer. Math. Soc. 14 (2001), no. 2, 279 – 296. · Zbl 1001.35079 [46] Ronald DiPerna and Pierre-Louis Lions, Solutions globales d’équations du type Vlasov-Poisson, C. R. Acad. Sci. Paris Sér. I Math. 307 (1988), no. 12, 655 – 658 (French, with English summary). · Zbl 0682.35022 [47] R. J. DiPerna and P.-L. Lions, Global weak solutions of Vlasov-Maxwell systems, Comm. Pure Appl. Math. 42 (1989), no. 6, 729 – 757. · Zbl 0698.35128 [48] R. J. DiPerna and P.-L. Lions, Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math. 98 (1989), no. 3, 511 – 547. · Zbl 0696.34049 [49] R. J. DiPerna and P.-L. Lions, On the Cauchy problem for Boltzmann equations: global existence and weak stability, Ann. of Math. (2) 130 (1989), no. 2, 321 – 366. · Zbl 0698.45010 [50] R. J. DiPerna, P.-L. Lions, and Y. Meyer, \?^{\?} regularity of velocity averages, Ann. Inst. H. Poincaré Anal. Non Linéaire 8 (1991), no. 3-4, 271 – 287 (English, with French summary). · Zbl 0763.35014 [51] J. Dolbeault, 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. · Zbl 1115.82316 [52] Bruno Dubroca and Jean-Luc Feugeas, Étude théorique et numérique d’une hiérarchie de modèles aux moments pour le transfert radiatif, C. R. Acad. Sci. Paris Sér. I Math. 329 (1999), no. 10, 915 – 920 (French, with English and French summaries). · Zbl 0940.65157 [53] Miguel Escobedo and Stephane Mischler, On a quantum Boltzmann equation for a gas of photons, J. Math. Pures Appl. (9) 80 (2001), no. 5, 471 – 515 (English, with English and French summaries). · Zbl 1134.82318 [54] Lawrence C. Evans, Partial differential equations, Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 1998. · Zbl 0902.35002 [55] Emmanuel Frénod and Eric Sonnendrücker, The finite Larmor radius approximation, SIAM J. Math. Anal. 32 (2001), no. 6, 1227 – 1247. · Zbl 0980.82030 [56] I. Gasser, P.-E. Jabin, and B. Perthame, Regularity and propagation of moments in some nonlinear Vlasov systems, Proc. Roy. Soc. Edinburgh Sect. A 130 (2000), no. 6, 1259 – 1273. · Zbl 0984.35102 [57] Patrick Gérard, Moyennisation et régularité deux-microlocale, Ann. Sci. École Norm. Sup. (4) 23 (1990), no. 1, 89 – 121 (French). · Zbl 0725.35003 [58] Patrick Gérard and François Golse, Averaging regularity results for PDEs under transversality assumptions, Comm. Pure Appl. Math. 45 (1992), no. 1, 1 – 26. · Zbl 0832.35020 [59] Patrick Gérard, Peter A. Markowich, Norbert J. Mauser, and Frédéric Poupaud, Homogenization limits and Wigner transforms, Comm. Pure Appl. Math. 50 (1997), no. 4, 323 – 379. , https://doi.org/10.1002/(SICI)1097-0312(199704)50:43.3.CO;2-Q · Zbl 0881.35099 [60] J. Ginibre and G. Velo, The global Cauchy problem for the nonlinear Schrödinger equation revisited, Ann. Inst. H. Poincaré Anal. Non Linéaire 2 (1985), no. 4, 309 – 327 (English, with French summary). · Zbl 0586.35042 [61] Robert T. Glassey, The Cauchy problem in kinetic theory, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1996. · Zbl 0858.76001 [62] Robert T. Glassey and Jack Schaeffer, The relativistic Vlasov-Maxwell system in 2D and 2.5D, Nonlinear wave equations (Providence, RI, 1998) Contemp. Math., vol. 263, Amer. Math. Soc., Providence, RI, 2000, pp. 61 – 69. · Zbl 0973.35129 [63] R. T. Glassey and W. A. Strauss, Asymptotic stability of the relativistic Maxwellian via fourteen moments, Transport Theory Statist. Phys. 24 (1995), no. 4-5, 657 – 678. · Zbl 0882.35123 [64] Robert T. Glassey and Walter A. Strauss, Decay of the linearized Boltzmann-Vlasov system, Transport Theory Statist. Phys. 28 (1999), no. 2, 135 – 156. · Zbl 0983.82018 [65] François Golse, Pierre-Louis Lions, Benoît Perthame, and Rémi Sentis, Regularity of the moments of the solution of a transport equation, J. Funct. Anal. 76 (1988), no. 1, 110 – 125. · Zbl 0652.47031 [66] François Golse, Benoît Perthame, and Rémi Sentis, Un résultat de compacité pour les équations de transport et application au calcul de la limite de la valeur propre principale d’un opérateur de transport, C. R. Acad. Sci. Paris Sér. I Math. 301 (1985), no. 7, 341 – 344 (French, with English summary). · Zbl 0591.45007 [67] François Golse and Laure Saint-Raymond, The Vlasov-Poisson system with strong magnetic field, J. Math. Pures Appl. (9) 78 (1999), no. 8, 791 – 817. · Zbl 0977.35108 [68] Golse, F.; Saint-Raymond, L. The Navier-Stokes limit of Boltzmann equation: convergence proof. Inventiones Mathematicae 155, no. 1 (2004), 81-161. · Zbl 1060.76101 [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] Thierry Goudon and Frederic Poupaud, Approximation by homogenization and diffusion of kinetic equations, Comm. Partial Differential Equations 26 (2001), no. 3-4, 537 – 569. · Zbl 0988.35023 [71] Emmanuel Grenier, Oscillations in quasineutral plasmas, Comm. Partial Differential Equations 21 (1996), no. 3-4, 363 – 394. · Zbl 0849.35107 [72] Yan Guo, Smooth irrotational flows in the large to the Euler-Poisson system in \?³$$^{+}$$\textonesuperior , Comm. Math. Phys. 195 (1998), no. 2, 249 – 265. , https://doi.org/10.1007/s002200050388 Yan Guo and Walter A. Strauss, Unstable BGK solitary waves and collisionless shocks, Comm. Math. Phys. 195 (1998), no. 2, 267 – 293. · Zbl 0944.35065 [73] Yan Guo and Walter A. Strauss, Magnetically created instability in a collisionless plasma, J. Math. Pures Appl. (9) 79 (2000), no. 10, 975 – 1009 (English, with English and French summaries). · Zbl 0985.76105 [74] K. Hamdache, Global existence and large time behaviour of solutions for the Vlasov-Stokes equations, Japan J. Indust. Appl. Math. 15 (1998), no. 1, 51 – 74. · Zbl 1306.76052 [75] E. Horst, On the classical solutions of the initial value problem for the unmodified nonlinear Vlasov equation. I. General theory, Math. Methods Appl. Sci. 3 (1981), no. 2, 229 – 248. , https://doi.org/10.1002/mma.1670030117 E. Horst, On the classical solutions of the initial value problem for the unmodified nonlinear Vlasov equation. II. Special cases, Math. Methods Appl. Sci. 4 (1982), no. 1, 19 – 32. · Zbl 0485.35079 [76] E. Horst and R. Hunze, Weak solutions of the initial value problem for the unmodified nonlinear Vlasov equation, Math. Methods Appl. Sci. 6 (1984), no. 2, 262 – 279. · Zbl 0556.35022 [77] R. Illner, A. Klar, H. Lange, A. Unterreiter, and R. Wegener, A kinetic model for vehicular traffic: existence of stationary solutions, J. Math. Anal. Appl. 237 (1999), no. 2, 622 – 643. · Zbl 0951.90013 [78] Reinhard Illner and Gerhard Rein, 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. , https://doi.org/10.1002/(SICI)1099-1476(19961125)19:173.3.CO;2-U · Zbl 0872.35087 [79] Pierre-Emmanuel Jabin, Macroscopic limit of Vlasov type equations with friction, Ann. Inst. H. Poincaré Anal. Non Linéaire 17 (2000), no. 5, 651 – 672 (English, with English and French summaries). · Zbl 0965.35013 [80] Pierre-Emmanuel Jabin and Benoit Perthame, Notes on mathematical problems on the dynamics of dispersed particles interacting through a fluid, Modeling in applied sciences, Model. Simul. Sci. Eng. Technol., Birkhäuser Boston, Boston, MA, 2000, pp. 111 – 147. · Zbl 0957.76087 [81] Pierre-Emmanuel Jabin and Benoît Perthame, Regularity in kinetic formulations via averaging lemmas, ESAIM Control Optim. Calc. Var. 8 (2002), 761 – 774. A tribute to J. L. Lions. · Zbl 1065.35185 [82] Jabin, P.-E.; Vega, L. A real space method for averaging lemmas. Preprint ENS-DMA 03-09 (2003). · Zbl 1082.35043 [83] Michael Junk, Domain of definition of Levermore’s five-moment system, J. Statist. Phys. 93 (1998), no. 5-6, 1143 – 1167. · Zbl 0952.82024 [84] Shuichi Kawashima, Akitaka Matsumura, and Takaaki Nishida, 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. · Zbl 0449.76053 [85] Markus Keel and Terence Tao, Endpoint Strichartz estimates, Amer. J. Math. 120 (1998), no. 5, 955 – 980. · Zbl 0922.35028 [86] C. David Levermore, Moment closure hierarchies for kinetic theories, J. Statist. Phys. 83 (1996), no. 5-6, 1021 – 1065. · Zbl 1081.82619 [87] P.-L. Lions, Compactness in Boltzmann’s equation via Fourier integral operators and applications. I, II, J. Math. Kyoto Univ. 34 (1994), no. 2, 391 – 427, 429 – 461. P.-L. Lions, Compactness in Boltzmann’s equation via Fourier integral operators and applications. III, J. Math. Kyoto Univ. 34 (1994), no. 3, 539 – 584. · Zbl 0831.35139 [88] Pierre-Louis Lions, Régularité optimale des moyennes en vitesses. II, C. R. Acad. Sci. Paris Sér. I Math. 326 (1998), no. 8, 945 – 948 (French, with English and French summaries). · Zbl 0922.35135 [89] P.-L. Lions and N. Masmoudi, 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. · Zbl 0987.76088 [90] P.-L. Lions and B. Perthame, Propagation of moments and regularity for the 3-dimensional Vlasov-Poisson system, Invent. Math. 105 (1991), no. 2, 415 – 430 (English, with French summary). · Zbl 0741.35061 [91] Pierre-Louis Lions and Benoît Perthame, Lemmes de moments, de moyenne et de dispersion, C. R. Acad. Sci. Paris Sér. I Math. 314 (1992), no. 11, 801 – 806 (French, with English summary). · Zbl 0761.35085 [92] P.-L. Lions, B. Perthame, and E. Tadmor, A kinetic formulation of multidimensional scalar conservation laws and related equations, J. Amer. Math. Soc. 7 (1994), no. 1, 169 – 191. · Zbl 0820.35094 [93] Liu, T.-P.; Yu, S.-H. Boltzmann equation: micro-macro decompositions and positivity or shock profiles. Preprint 2002. [94] P. A. Markowich, C. A. Ringhofer, and C. Schmeiser, Semiconductor equations, Springer-Verlag, Vienna, 1990. · Zbl 0765.35001 [95] Stéphane Mischler, On the initial boundary value problem for the Vlasov-Poisson-Boltzmann system, Comm. Math. Phys. 210 (2000), no. 2, 447 – 466. · Zbl 0983.45007 [96] Mischler, S. On weak-weak convergences and some applications to the initial boundary value problem for kinetic equations. Preprint (2001). [97] Stéphane Mischler and Bernst Wennberg, On the spatially homogeneous Boltzmann equation, Ann. Inst. H. Poincaré Anal. Non Linéaire 16 (1999), no. 4, 467 – 501 (English, with English and French summaries). · Zbl 0946.35075 [98] Juan Nieto, Frédéric Poupaud, and Juan Soler, High-field limit for the Vlasov-Poisson-Fokker-Planck system, Arch. Ration. Mech. Anal. 158 (2001), no. 1, 29 – 59. · Zbl 1038.82068 [99] H. G. Othmer, S. R. Dunbar, and W. Alt, Models of dispersal in biological systems, J. Math. Biol. 26 (1988), no. 3, 263 – 298. · Zbl 0713.92018 [100] George Papanicolaou and Leonid Ryzhik, Waves and transport, Hyperbolic equations and frequency interactions (Park City, UT, 1995) IAS/Park City Math. Ser., vol. 5, Amer. Math. Soc., Providence, RI, 1999, pp. 305 – 382. · Zbl 0930.35172 [101] B. Perthame, Time decay, propagation of low moments and dispersive effects for kinetic equations, Comm. Partial Differential Equations 21 (1996), no. 3-4, 659 – 686 (English, with English and French summaries). · Zbl 0852.35139 [102] Perthame, B. Kinetic formulation of conservation laws. Oxford Univ. Press, Series in Math. and Appl. 21 (2002). · Zbl 1030.35002 [103] B. Perthame and P. E. Souganidis, A limiting case for velocity averaging, Ann. Sci. École Norm. Sup. (4) 31 (1998), no. 4, 591 – 598 (English, with English and French summaries). · Zbl 0956.45010 [104] Benoit Perthame and Luis Vega, Morrey-Campanato estimates for Helmholtz equations, J. Funct. Anal. 164 (1999), no. 2, 340 – 355. · Zbl 0932.35048 [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] F. Poupaud, Runaway phenomena and fluid approximation under high fields in semiconductor kinetic theory, Z. Angew. Math. Mech. 72 (1992), no. 8, 359 – 372 (English, with English, German and Russian summaries). · Zbl 0785.76067 [107] K. Pfaffelmoser, Global classical solutions of the Vlasov-Poisson system in three dimensions for general initial data, J. Differential Equations 95 (1992), no. 2, 281 – 303. · Zbl 0810.35089 [108] Giovanni Russo and Peter Smereka, Kinetic theory for bubbly flow. I. Collisionless case, SIAM J. Appl. Math. 56 (1996), no. 2, 327 – 357. , https://doi.org/10.1137/S0036139993260563 Giovanni Russo and Peter Smereka, Kinetic theory for bubbly flow. II. Fluid dynamic limit, SIAM J. Appl. Math. 56 (1996), no. 2, 358 – 371. · Zbl 0857.76091 [109] L. Saint-Raymond, The gyrokinetic approximation for the Vlasov-Poisson system, Math. Models Methods Appl. Sci. 10 (2000), no. 9, 1305 – 1332. · Zbl 0965.35130 [110] Jack Schaeffer, Global existence of smooth solutions to the Vlasov-Poisson system in three dimensions, Comm. Partial Differential Equations 16 (1991), no. 8-9, 1313 – 1335. · Zbl 0746.35050 [111] Yoshio Sone, Kinetic theory and fluid dynamics, Modeling and Simulation in Science, Engineering and Technology, Birkhäuser Boston, Inc., Boston, MA, 2002. · Zbl 1021.76002 [112] Giuseppe Toscani, One-dimensional kinetic models of granular flows, M2AN Math. Model. Numer. Anal. 34 (2000), no. 6, 1277 – 1291. · Zbl 0981.76098 [113] C. Truesdell and R. G. Muncaster, Fundamentals of Maxwell’s kinetic theory of a simple monatomic gas, Pure and Applied Mathematics, vol. 83, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1980. Treated as a branch of rational mechanics. [114] Seiji Ukai, The Boltzmann-Grad limit and Cauchy-Kovalevskaya theorem, Japan J. Indust. Appl. Math. 18 (2001), no. 2, 383 – 392. Recent topics in mathematics moving toward science and engineering. · Zbl 0981.35004 [115] Seiji Ukai and Kiyoshi Asano, 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. · Zbl 0538.76070 [116] Cédric Villani, A review of mathematical topics in collisional kinetic theory, Handbook of mathematical fluid dynamics, Vol. I, North-Holland, Amsterdam, 2002, pp. 71 – 305. · Zbl 1170.82369 [117] Villani, C. Cercignani’s conjecture is sometimes true and always almost true. Comm. Math. Phys. 234 (2003), no. 3, 455-490. · Zbl 1041.82018 [118] Michael Westdickenberg, Some new velocity averaging results, SIAM J. Math. Anal. 33 (2002), no. 5, 1007 – 1032. · Zbl 1067.35021
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.