From the boltzmann equation to generalized kinetic models in applied sciences

Mathematical and Computer Modelling

Volumn 26, Issue 7, 1997, Pages 43-76

  Bellomo, N ^a   Schiavo, M Lo ^b  

^a POLITECNICO DI TORINO   (Italy)

^b SAPIENZA UNIVERSITY OF ROME   (Italy)

Author keywords

Boltzmann equation; Evolution problems; Kinetic models; Nonlinear systems

Indexed keywords

FUNCTIONS; INTEGRODIFFERENTIAL EQUATIONS; NONLINEAR EQUATIONS; NUMERICAL ANALYSIS; STATISTICAL MECHANICS;

APPLIED SCIENCES; BOLTZMANN EQUATION; EVOLUTION PROBLEM; GENERALIZED KINETIC MODELS;

MATHEMATICAL MODELS;

EID: 0031259318     PISSN: 08957177     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0895-7177(97)00185-4     Document Type: Article

References (88)

1. C. Cercignani, Theory and Application of the Boltzmann Equation, Springer, (1988).
2. C. Cercignani, R. Illner and M. Pulvirenti, Theory and Application of the Boltzmann Equation, Springer, (1993).
3. C. Truesdell and R. Muncaster, Fundamentals of Maxwell's Kinetic Theory of a Simple Monoatomic Gas, Academic Press, (1980).
4. N. Bellomo, A. Palczewski and G. Toscani, Mathematical Topics in Nonlinear Kinetic Theory, World Sci., (1988).
5. A. Chorin and J. Marsden, Mathematical Introduction to Fluid Dynamics, Springer, (1979).
6. N. Bellomo, M. Lachowicz, J. Polewczak and G. Toscani, Mathematical Topics in Nonlinear Kinetic Theory II: The Enskog Equation, World Sci., (1991).
7. R. Glassey, The Cauchy Problem in Kinetic Theory, SIAM Publ., (1996).
8. N. Bellomo, P. Le Tallec and B. Perthame, On the solution of the nonlinear Boltzmann equation, ASME Review 48, 777-794, (1995).
9. H. Neunzert and J. Struckmeier, Particle methods for the Boltzmann equation, In Acta Numerica 1995, pp. 417-458, Cambridge Univ. Press, Cambridge, (1995).
10. N. Bellomo, Editor, Lecture Notes on the Mathematical Theory of the Boltzmann Equation, World Sci., (1995).
11. L. Arlotti and N. Bellomo, On the Cauchy problem for the nonlinear Boltzmann equation, In Lecture Notes on the Mathematical Theory of the Boltzmann Equation, pp. 1-64, World Sci., (1995).
12. R. DiPerna and P.L. Lions, On the Cauchy problem for the Boltzmann equation: Global existence and weak stability results, Annals Math. 130, 1189-1214, (1990).
13. R. DiPerna and P.L. Lions, Global solutions of the Boltzmann and the entropy inequality, Arch. Rat. Mech. Anal. 114, 47-55, (1991).
14. P.L. Lions, Compactness in Boltzmann equation via Fourier integrals operators and applications, J. Math. Kyoto Univ. 34, I: 391-427, II: 429-460, III: 539-584, (1994).
15. P.L. Lions, Compactness in Boltzmann equation via Fourier integrals operators and applications, J. Math. Kyoto Univ. 34, I: 391-427, II: 429-460, III: 539-584, (1994).
16. P.L. Lions, Compactness in Boltzmann equation via Fourier integrals operators and applications, J. Math. Kyoto Univ. 34, I: 391-427, II: 429-460, III: 539-584, (1994).
17. R. Glassey, Global existence of the relativistic Vlasov-Maxwell system with nearly neutral initial data, Comm. Math. Phys. 119, 353-384, (1988).
18. R. Glassey, On the one and one-half dimensional relativistic Vlasov-Maxwell system, Math. Meth. Appl. Sci. 113, 169-179, (1990).
19. L. Arkeryd and C. Cercignani, A global existence theorem for the initial-boundary value problem for the Boltzmann equation when boundaries are not isothermal, Arch. Rat. Mech. Anal. 125, 271-287, (1993).
20. L. Arkeryd and N. Maslova, On diffuse reflection at the boundary for the Boltzmann equation and related equations, J. Statist. Phys. 77, 271-287, (1994).
21. L. Arkeryd and A. Nouri, Asymptotics of the Boltzmann equation with diffuse reflection boundary conditions, Rapporte Interne Univ. de Nice 406, (1994).
22. N. Bellomo and J. Polewczak, On the generalized Boltzmann equation, global existence and exponential trend to equilibrium, Transp. Theory Statist. Phys. 26, (1997) (to appear).
23. E. Jager and L. Segel, On the distribution of dominance in a population of interacting anonymous organisms, SIAM J. Appl. Math. 52, 1442-1468, (1992).
24. P. Hogeweg and P. Hesper, The ontogeny of the interaction structure in bumble bee colonies, Behav. Ecol. Sociobiol. 12, 271-183, (1983).
25. L. Arlotti and N. Bellomo, Population dynamics with stochastic interactions, Transp. Theory Statist. Phys. 24, 431-443, (1995).
26. L. Arlotti and N. Bellomo, Solution of a new class of nonlinear kinetic models of population dynamics, Appl. Math. Letters 9 (2), 65-70, (1996).
27. F. Hoppenstead, Mathematical theory of populations: Demographic, genetics and epidemics, SIAM Conf. Series 2, (1975).
28. A. Canada and A. Zertiti, Method of upper and lower solutions for nonlinear delay integral equations modelling epidemics and population growth, Math. Models Meth. Appl. Sci. 4, 107-120, (1994).
29. L. Preziosi, From population dynamics to modelling the competition between tumors and immune system, Comp. Math. Modelling 23, 135-152, (1996).
30. G. Forni, R. Foa, A. Santoni and L. Frati, Editors, Cytokine Induced Tumor Immunogeneticity, Academic Press, (1994).
31. N. Bellomo and G. Forni, Dynamics of tumor interaction with the host immune system, Mathl. Comput. Modelling 20 (1), 107-122, (1994).
32. N. Bellomo, G. Forni and L. Preziosi, On a kinetic (cellular) theory for the competition between tumors and host-immune system, J. Biol. Systems 4, 479-502, (1996).
33. N. Bellomo, B. Firmani, L. Guerri and L. Preziosi, On a kinetic theory of Cytokine mediated interaction between tumors and immune system, ARI Journal 1, (1997) (to appear).
34. Adam J. and N. Bellomo, Editors, A Survey of Models on Tumor Immune Systems Dynamics, Birkhäuser, (1996).
35. L. Arlotti and M. Lachowicz, Qualitative analysis of a nonlinear integro-differential equation modelling tumor-host dynamics, Mathl. Comput. Modelling 23 (6), 11-30, (1996).
36. M. Lo Schiavo, Discrete kinetic cellular models of tumors immune system interactions, Math. Models Meth. Appl. Sci. 6, 1187-1210, (1996).
37. R.J. De Boer, L. Segel and S. Perelson, Pattern formation in one or two dimensional shape-space models of the immune system, J. Theor. Biol. 155, 295-333, (1992).
38. R.J. De Boer, M.C. Boerlijst, B. Sulzer and S. Perelson, A new bell shaped functions for idiotypic interactions based on cross linking, Bull. Math. Biol. 58, 285-312, (1996).
39. L.A. Segel and A.S. Perelson, Computations in shape space: A new approach to immune network theory, In Theoretical Immunology, Part Two, SFI Studies in the Science of Complexity, (Edited by A.S. Perelson), pp. 321-342, Addison-Wesley, (1988).
40. N. Bellomo, M. Lachowicz and J. Polewczak, On the evolution of the generalized shape in immunology: Qualitative analysis of the solutions to Segel-Perelson model, Appl. Math. Letters 10 (4), 53-58, (1997).
41. I. Prigogine and R. Herman, Kinetic Theory of Vehicular Traffic, Elsevier, (1971).
42. M. Lampis, On the kinetic theory of traffic flow in the case of a non negligible number of queuing vehicles, Transport. Science 12, 16-28, (1978).
43. S. Paveri Fontana, On Boltzmann like treatment of vehicular traffic, Transport. Science 9, 225-235, (1975).
44. A. Belleni Morante, Nonlinear kinetic theory of vehicular traffic, J. Math. Anal. Appl. 47, 443-457, (1974).
45. P. Nelson, A kinetic model of vehicular traffic associated to bimodal distribution, Transp. Theory Statist. Phys. 24, 383-39, (1995).
46. A. Klar, R.D. Küne and R. Wegner, Mathematical models for vehicular traffic, In Surveys Math. Ind., Int. Report 95-158, Arbeitsgruppe Technomathematik, Univ. Kaiserslautern, (1995) (to appear).
47. A. Klar and R. Wegner, Enskog-like models for vehicular traffic, J. Statist. Phys. 87, 91-114, (1996).
48. R. Wegener and A. Klar, A kinetic model for vehicular traffic derived from a stochastic microscopic model, Transp. Theory Statist. Phys. 25, 785-798, (1996).
49. L. Preziosi and L. de Socio, A nonlinear invverse phase transiton problem for the heat equation, Math. Models Meth. Appl. Sci. 1, 1187-1210, (1991).
50. N. Bellomo and L. Preziosi, Modelling Mathematical Methods and Scientific Computation, CRC Press, (1995).
51. W.H. White, Global existence theorem for Smoluchowski's coagulation equation, Proc. Amer. Math. Soc. 80, 273-276, (1980).
52. I.W. Stewart, A global existence theorem for the general coagulation fragmentation equation with unbounded kernels, Math. Meth. Appl. Sci. 11, 627-648, (1989).
53. I.W. Stewart, On the coagulation fragmentation equation, J. Appl. Math. Phys. 41, 917-924, (1990).
54. I.W. Stewart, Density conservation for a coagulation equation, J. Appl. Math. Phys. 42, 746-756, (1991).
55. I.W. Stewart and P.B. Dubovskii, Approach to equilibrium for the coagulation fragmentation equation via a Lyapunov functional, Math. Meth. Appl. Sci. 19, 171-185, (1996).
56. A.S. Ackleh, B.G. Fitzpatrick and T.G. Hallam, Approximation and parameter estimation problems for algal aggregation models, Math. Models Meth. Appl. Sci. 4, 291-311, (1994).
57. J. Collet and F. Popoud, Existence of solutions to coagulation-fragmentation systems with diffusion, Transp. Theory Statist. Phys. 25, 503-513, (1996).
58. D. Wrzosek, Existence of solutions for the discrete coagulation fragmentation model with diffusion, Int. Report 96-04 (16) of the Inst. of Appl. Mathematics University of Warsaw (1996).
59. M. Slemrod and A. Qi, A discrete velocity coagulation fragmentation model, Math. Models Meth. Appl. Sci. 5, 619-640, (1995).
60. M. Esteban and B. Perthame, Solutions globale de l'équation d'Enskog modifiée avec collisions élastique ou inélastique, Comp. Rend. Acad. Sci. Paris 309, 897-902, (1989).
61. M. Esteban and B. Perthame, On the modified Enskog equation for elastic and inelastic collisions. Models with Spin., Ann. Inst. Poincaré 8, 289-308, (1991).
62. S. Araki and S. Tremaine, The dynamics of dense particle disks, Icarus 65, 83-109, (1986).
63. S. Araki, The dynamics of dense particle disks - Effects of Spin degrees of freedom, Icarus 65, 182-198, (1988).
64. N. Bellomo, M. Esteban and M. Lachowicz, Nonlinear kinetic equations with dissipative collisions, Appl. Math. Lett. 8 (5), 47-52, (1995).
65. G. Russo and P. Smereka, Kinetic theory of bubbly flow I: Collisionless case, SIAM J. Appl. Math. 56, 327-357, (1996).
66. G. Russo and P. Smereka, Kinetic theory of bubbly flow II: Collisionless case, SIAM J. Appl. Math. 56, 357-371, (1996).
67. M. Lachowicz, Asymptotic analysis of nonlinear kinetic equations - The hydrodynamic limit, In Lectures Notes on the Mathematical Theory of the Boltzmann Equation, (Edited by N. Bellomo), pp. 65-148, World Sci., (1995).
68. P. Markowich, C. Ringhofer and C. Schmeiser, Semiconductor Equations, Springer, (1989).
69. A. Kersch and J. Morokoff, Transport Simulation in Microelectronics, Birkhäuser, (1995).
70. N. Ben Abdallah and P. Degond, On a hierarchy of macroscopic models for semiconductors, J. Math. Phys. 37, 3306-3333, (1966).
71. N. Ben Abdallah and P. Degond, An energy-transport model for semiconductors derived from the Boltzmann equation, J. Stat. Phys. 84, 205-231, (1966).
72. G. Kaniadakis and P. Quarati, Classical models of bosons and fermions, Phys. Review E 49, 5103-5110, (1994).
73. G. Kaniadakis, Classical models of intermediate statistics, Phys. Review E 49, 5111-5116, (1994).
74. B. Niclot, P. Degond and F. Poupaud, Deterministic particle simulation of the Boltzmann transport equation of semiconductors, J. Comp. Phys. 78, 313-349, (1990).
75. P. Degond and F. Guyot-Delaurens, Particle simulation of the semiconductor Boltzmann equation for one-dimensional inhomogeneous structures, J. Comp. Phys. 90, 65-97, (1990).
76. F. Popaud, Mathematical theory of kinetic equations for transport modelling in semiconductors, In Kinetic Theory and Computing, (Edited by B. Perthame), World Sci., (1994).
77. C. Buet, A discrete velocity scheme for the Boltzmann operator of rarefied gas-dynamics, Transp. Theory Statist. Phys. 25, 33-60, (1996).
78. C. Buet, Conservative and entropy schemes for Boltzmann collision operator of polyatomic gases, Math. Models Meth. Appl. Sci. 7, 165-192, (1997).
79. L. Preziosi and E. Longo, The semicontinuous Boltzmann equation, Math. Models Meth. Appl. Sci. 3, 65-94, (1993).
80. L. Preziosi and E. Longo, On a conservative polar discretization of the Boltzmann equation, Japan J. Ind. Appl. Math. (to appear).
81. L. Arlotti and M. Lachowicz, On the hydrodynamic limit for quantum kinetic equations, Comp. Rend. Acad. Science Paris 323, 101-106, (1996).
82. M. Lachowicz, From kinetic to Navier-Stokes type equations, Appl. Math. Lett. 10 (5), 19-24, (1997).
83. F. Fuchs and F. Poupaud, Asymptotical and numerical analysis of degeneracy effects on the drift-diffusion equations for semiconductors, Math. Models Meth. Appl. Sci. 5, 1093-1111, (1995).
84. H. Lange, V. Toomire and P. Zweifel, A survey of Wiegner-Poisson problems, In Operator Theory: Advances and Applications, Volume 70, pp. 239-256, Birkhäuser, (1993).
85. H. Lange, H. Toomire and P. Zweifel, Time-dependent dissipation in nonlinear Schrödinger equation, J. Math. Phys. 36, 1274-1283, (1995).
86. H. Lange and P. Zweifel, Quantum BGK models for the Wiegner-Poisson equation, Transp. Theory Statist. Phys. 25, 713-722, (1996).
87. P. Zweifel, The Wiegner transform and the Wiegner-Poisson system, Transp. Theory Statist. Phys. 22, 459-484, (1993).
88. P. Zweifel, The Wiegner-Poisson system: A quantum transport equation, Reports Math. Phys. 33, 317-323, (1993).