From the Boltzmann equation to generalized kinetic models in applied sciences. (English) Zbl 0897.76082
Summary: This paper deals with kinetic-type (Boltzmann) modelling and with the analysis of mathematical problems related to the application of models. We start from a concise report on Boltzmann equation and go to the analysis of models linked to this fundamental equation of nonequilibrium statistical mechanics. Our aim is to provide a guideline towards modelling systems of interest in applied sciences by means of kinetic-type equations.
MSC:
| 76P05 | Rarefied gas flows, Boltzmann equation in fluid mechanics |
| 00A71 | General theory of mathematical modeling |
References:
| [1] | Cercignani, C., Theory and Application of the Boltzmann Equation (1988), Springer · Zbl 0646.76001 |
| [2] | Cercignani, C.; Illner, R.; Pulvirenti, M., Theory and Application of the Boltzmann Equation (1993), Springer |
| [3] | Truesdell, C.; Muncaster, R., Fundamentals of Maxwell’s Kinetic Theory of a Simple Monoatomic Gas (1980), Academic Press · Zbl 1515.80001 |
| [4] | Bellomo, N.; Palczewski, A.; Toscani, G., Mathematical Topics in Nonlinear Kinetic Theory (1988), World Sci · Zbl 0702.76005 |
| [5] | Chorin, A.; Marsden, J., Mathematical Introduction to Fluid Dynamics (1979), Springer · Zbl 0417.76002 |
| [6] | Bellomo, N.; Lachowicz, M.; Polewczak, J.; Toscani, G., Mathematical Topics in Nonlinear Kinetic Theory II: The Enskog Equation (1991), World Sci · Zbl 0733.76061 |
| [7] | Glassey, R., The Cauchy Problem in Kinetic Theory (1996), SIAM Publ · Zbl 0858.76001 |
| [8] | Bellomo, N.; Le Tallec, P.; Perthame, B., On the solution of the nonlinear Boltzmann equation, ASME Review, 48, 777-794 (1995) |
| [9] | Neunzert, H.; Struckmeier, J., Particle methods for the Boltzmann equation, (Acta Numerica 1995 (1995), Cambridge Univ. Press: Cambridge Univ. Press Cambridge), 417-458 · Zbl 0832.76086 |
| [10] | (Bellomo, N., Lecture Notes on the Mathematical Theory of the Boltzmann Equation (1995), World Sci) · Zbl 0871.76074 |
| [11] | Arlotti, L.; Bellomo, N., On the Cauchy problem for the nonlinear Boltzmann equation, (Lecture Notes on the Mathematical Theory of the Boltzmann Equation (1995), World Sci), 1-64 |
| [12] | DiPerna, R.; Lions, P. L., On the Cauchy problem for the Boltzmann equation: Global existence and weak stability results, Annals Math., 130, 1189-1214 (1990) |
| [13] | DiPerna, R.; Lions, P. L., Global solutions of the Boltzmann and the entropy inequality, Arch. Rat. Mech. Anal., 114, 47-55 (1991) · Zbl 0724.45011 |
| [14] | Lions, P. L., Compactness in Boltzmann equation via Fourier integrals operators and applications, J. Math. Kyoto Univ., 34, III, 539-584 (1994) · Zbl 0884.35124 |
| [15] | Glassey, R., Global existence of the relativistic Vlasov-Maxwell system with nearly neutral initial data, Comm. Math. Phys., 119, 353-384 (1988) · Zbl 0673.35070 |
| [16] | Glassey, R., On the one and one-half dimensional relativistic Vlasov-Maxwell system, Math. Meth. Appl. Sci., 113, 169-179 (1990) · Zbl 0722.35053 |
| [17] | Arkeryd, L.; Cercignani, C., 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) · Zbl 0789.76075 |
| [18] | Arkeryd, L.; Maslova, N., On diffuse reflection at the boundary for the Boltzmann equation and related equations, J. Statist. Phys., 77, 271-287 (1994) · Zbl 0839.76073 |
| [19] | Arkeryd, L.; Nouri, A., Asymptotics of the Boltzmann equation with diffuse reflection boundary conditions, Rapporte Interne Univ. de Nice, 406 (1994) · Zbl 0877.76063 |
| [20] | Bellomo, N.; Polewczak, J., On the generalized Boltzmann equation, global existence and exponential trend to equilibrium, Transp. Theory Statist. Phys., 26 (1997), (to appear) · Zbl 0903.35053 |
| [21] | Jager, E.; Segel, L., On the distribution of dominance in a population of interacting anonymous organisms, SIAM J. Appl Math., 52, 1442-1468 (1992) · Zbl 0759.92011 |
| [22] | Hogeweg, P.; Hesper, P., The ontogeny of the interaction structure in bumble bee colonies, Behav. Ecol. Sociobiol., 12, 183-271 (1983) |
| [23] | Arlotti, L.; Bellomo, N., Population dynamics with stochastic interactions, Transp. Theory Statist. Phys., 24, 431-443 (1995) · Zbl 0838.92018 |
| [24] | Arlotti, L.; Bellomo, N., Solution of a new class of nonlinear kinetic models of population dynamics, Appl. Math. Letters, 9, 2, 65-70 (1996) · Zbl 0853.35050 |
| [25] | Hoppenstead, F., Mathematical theory of populations: Demographic, genetics and epidemics, (SIAM Conf. Series, 2 (1975)) · Zbl 0304.92012 |
| [26] | Canada, A.; Zertiti, A., 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) · Zbl 0817.92013 |
| [27] | Preziosi, L., From population dynamics to modelling the competition between tumors and immune system, Comp. Math. Modelling, 23, 135-152 (1996) · Zbl 0897.92016 |
| [28] | (Forni, G.; Foa, R.; Santoni, A.; Frati, L., Cytokine Induced Tumor Immunogeneticity (1994), Academic Press) |
| [29] | Bellomo, N.; Forni, G., Dynamics of tumor interaction with the host immune system, Mathl. Comput. Modelling, 20, 1, 107-122 (1994) · Zbl 0811.92014 |
| [30] | Bellomo, N.; Forni, G.; Preziosi, L., On a kinetic (cellular) theory for the competition between tumors and host-immune system, J. Biol. Systems, 4, 479-502 (1996) |
| [31] | Bellomo, N.; Firmani, B.; Guerri, L.; Preziosi, L., On a kinetic theory of Cytokine mediated interaction between tumors and immune system, ARI Journal, 1 (1997), (to appear) |
| [32] | (Adam, J.; Bellomo, N., A Survey of Models on Tumor Immune Systems Dynamics (1996), Birkhäuser) |
| [33] | Arlotti, L.; Lachowicz, M., Qualitative analysis of a nonlinear integro-differential equation modelling tumor-host dynamics, Mathl. Comput. Modelling, 23, 6, 11-30 (1996) · Zbl 0859.92011 |
| [34] | Lo Schiavo, M., Discrete kinetic cellular models of tumors immune system interactions, Math. Models Meth. Appl. Sci., 6, 1187-1210 (1996) · Zbl 0874.92022 |
| [35] | De Boer, R. J.; Segel, L.; Perelson, S., Pattern formation in one or two dimensional shape-space models of the immune system, J. Theor. Biol., 155, 295-333 (1992) |
| [36] | De Boer, R. J.; Boerlijst, M. C.; Sulzer, B.; Perelson, S., A new bell shaped functions for idiotypic interactions based on cross linking, Bull. Math. Biol., 58, 285-312 (1996) · Zbl 0853.92006 |
| [37] | Segel, L. A.; Perelson, A. S., Computations in shape space: A new approach to immune network theory, (Perelson, A. S., Theoretical Immunology, Part Two, SFI Studies in the Science of Complexity (1988), Addison-Wesley), 321-342 |
| [38] | Bellomo, N.; Lachowicz, M.; Polewczak, J., 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) · Zbl 0898.92017 |
| [39] | Prigogine, I.; Herman, R., Kinetic Theory of Vehicular Traffic (1971), Elsevier · Zbl 0226.90011 |
| [40] | Lampis, M., On the kinetic theory of traffic flow in the case of a non negligible number of queuing vehicles, Transport. Science, 12, 16-28 (1978) |
| [41] | Paveri Fontana, S., On Boltzmann like treatment of vehicular traffic, Transport. Science, 9, 225-235 (1975) |
| [42] | Belleni Morante, A., Nonlinear kinetic theory of vehicular traffic, J. Math. Anal. Appl., 47, 443-457 (1974) · Zbl 0297.45009 |
| [43] | Nelson, P., A kinetic model of vehicular traffic associated to bimodal distribution, Transp. Theory Statist. Phys., 24, 39-383 (1995) · Zbl 0817.90028 |
| [44] | Klar, A.; Küne, R. D.; Wegner, R., Mathematical models for vehicular traffic, (Surveys Math. Ind., Int. Report 95-158 (1995), Arbeitsgruppe Technomathematik, Univ. Kaiserslautern), (to appear) |
| [45] | Klar, A.; Wegner, R., Enskog-like models for vehicular traffic, J. Statist. Phys., 87, 91-114 (1996) · Zbl 0917.90124 |
| [46] | Wegener, R.; Klar, A., A kinetic model for vehicular traffic derived from a stochastic microscopic model, Transp. Theory Statist. Phys., 25, 785-798 (1996) · Zbl 1467.82077 |
| [47] | Preziosi, L.; de Socio, L., A nonlinear invverse phase transiton problem for the heat equation, Math. Models Meth. Appl. Sci., 1, 1187-1210 (1991) |
| [48] | Bellomo, N.; Preziosi, L., Modelling Mathematical Methods and Scientific Computation (1995), CRC Press · Zbl 0871.65001 |
| [49] | White, W. H., Global existence theorem for Smoluchowski’s coagulation equation, (Proc. Amer. Math. Soc., 80 (1980)), 273-276 · Zbl 0442.34003 |
| [50] | Stewart, I. W., A global existence theorem for the general coagulation fragmentation equation with unbounded kernels, Math. Meth. Appl. Sci., 11, 627-648 (1989) · Zbl 0683.45006 |
| [51] | Stewart, I. W., On the coagulation fragmentation equation, J. Appl. Math. Phys., 41, 917-924 (1990) · Zbl 0734.45009 |
| [52] | Stewart, I. W., Density conservation for a coagulation equation, J. Appl. Math. Phys., 42, 746-756 (1991) · Zbl 0752.45016 |
| [53] | Stewart, I. W.; Dubovskii, P. B., Approach to equilibrium for the coagulation fragmentation equation via a Lyapunov functional, Math. Meth. Appl. Sci., 19, 171-185 (1996) · Zbl 0840.45010 |
| [54] | Ackleh, A. S.; Fitzpatrick, B. G.; Hallam, T. G., Approximation and parameter estimation problems for algal aggregation models, Math. Models Meth. Appl. Sci., 4, 291-311 (1994) · Zbl 0823.92029 |
| [55] | Collet, J.; Popoud, F., Existence of solutions to coagulation-fragmentation systems with diffusion, Transp. Theory Statist. Phys., 25, 503-513 (1996) · Zbl 0870.35117 |
| [56] | Wrzosek, D., 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) |
| [57] | Slemrod, M.; Qi, A., A discrete velocity coagulation fragmentation model, Math. Models Meth. Appl. Sci., 5, 619-640 (1995) · Zbl 0833.76067 |
| [58] | Esteban, M.; Perthame, B., Solutions globale de l’équation d’Enskog modifiée avec collisions élastique ou inélastique, Comp. Rend. Acad. Sci. Paris, 309, 897-902 (1989) · Zbl 0706.35133 |
| [59] | Esteban, M.; Perthame, B., On the modified Enskog equation for elastic and inelastic collisions. Models with Spin, Ann. Inst. Poincaré, 8, 289-308 (1991) · Zbl 0850.70141 |
| [60] | Araki, S.; Tremaine, S., The dynamics of dense particle disks, Icarus, 65, 83-109 (1986) |
| [61] | Araki, S., The dynamics of dense particle disks—Effects of Spin degrees of freedom, Icarus, 65, 182-198 (1988) |
| [62] | Bellomo, N.; Esteban, M.; Lachowicz, M., Nonlinear kinetic equations with dissipative collisions, Appl. Math. Lett., 8, 5, 47-52 (1995) · Zbl 0835.76091 |
| [63] | Russo, G.; Smereka, P., Kinetic theory of bubbly flow I: Collisionless case, SIAM J. Appl. Math., 56, 327-357 (1996) · Zbl 0857.76090 |
| [64] | Russo, G.; Smereka, P., Kinetic theory of bubbly flow II: Collisionless case, SIAM J. Appl. Math., 56, 357-371 (1996) · Zbl 0857.76090 |
| [65] | Lachowicz, M., Asymptotic analysis of nonlinear kinetic equations—The hydrodynamic limit, (Bellomo, N., Lectures Notes on the Mathematical Theory of the Boltzmann Equation (1995), World Sci), 65-148 |
| [66] | Markowich, P.; Ringhofer, C.; Schmeiser, C., Semiconductor Equations (1989), Springer · Zbl 0667.65098 |
| [67] | Kersch, A.; Morokoff, J., Transport Simulation in Microelectronics (1995), Birkhäuser · Zbl 0828.76001 |
| [68] | Ben Abdallah, N.; Degond, P., On a hierarchy of macroscopic models for semiconductors, J. Math. Phys., 37, 3306-3333 (1966) · Zbl 0868.45006 |
| [69] | Ben Abdallah, N.; Degond, P., An energy-transport model for semiconductors derived from the Boltzmann equation, J. Stat. Phys., 84, 205-231 (1966) · Zbl 1081.82610 |
| [70] | Kaniadakis, G.; Quarati, P., Classical models of bosons and fermions, Phys. Review E, 49, 5103-5110 (1994) |
| [71] | Kaniadakis, G., Classical models of intermediate statistics, Phys. Review E, 49, 5111-5116 (1994) |
| [72] | Niclot, B.; Degond, P.; Poupaud, F., Deterministic particle simulation of the Boltzmann transport equation of semiconductors, J. Comp. Phys., 78, 313-349 (1990) · Zbl 0662.65126 |
| [73] | Degond, P.; Guyot-Delaurens, F., Particle simulation of the semiconductor Boltzmann equation for one-dimensional inhomogeneous structures, J. Comp. Phys., 90, 65-97 (1990) · Zbl 0731.65121 |
| [74] | Popaud, F., Mathematical theory of kinetic equations for transport modelling in semiconductors, (Perthame, B., Kinetic Theory and Computing (1994), World Sci) · Zbl 0863.76075 |
| [75] | Buet, C., A discrete velocity scheme for the Boltzmann operator of rarefied gas-dynamics, Transp. Theory Statist. Phys., 25, 33-60 (1996) · Zbl 0857.76079 |
| [76] | Buet, C., Conservative and entropy schemes for Boltzmann collision operator of polyatomic gases, Math. Models Meth. Appl. Sci., 7, 165-192 (1997) · Zbl 0872.76085 |
| [77] | Preziosi, L.; Longo, E., The semicontinuous Boltzmann equation, Math. Models Meth. Appl. Sci., 3, 65-94 (1993) · Zbl 0770.76057 |
| [78] | L. Preziosi and E. Longo, On a conservative polar discretization of the Boltzmann equation, Japan J. Ind. Appl. Math, (to appear).; L. Preziosi and E. Longo, On a conservative polar discretization of the Boltzmann equation, Japan J. Ind. Appl. Math, (to appear). · Zbl 0895.76070 |
| [79] | Arlotti, L.; Lachowicz, M., On the hydrodynamic limit for quantum kinetic equations, Comp. Rend. Acad. Science Paris, 323, 101-106 (1996) · Zbl 0856.76071 |
| [80] | Lachowicz, M., From kinetic to Navier-Stokes type equations, Appl. Math. Lett., 10, 5, 19-24 (1997) · Zbl 0897.76004 |
| [81] | Fuchs, F.; Poupaud, F., Asymptotical and numerical analysis of degeneracy effects on the drift-diffusion equations for semiconductors, Math. Models Meth. Appl. Sci., 5, 1093-1111 (1995) · Zbl 0847.76076 |
| [82] | Lange, H.; Toomire, V.; Zweifel, P., A survey of Wiegner-Poisson problems, (Operator Theory: Advances and Applications, Volume 70 (1993), Birkhäuser), 239-256 · Zbl 0826.35104 |
| [83] | Lange, H.; Toomire, H.; Zweifel, P., Time-dependent dissipation in nonlinear Schrödinger equation, J. Math. Phys., 36, 1274-1283 (1995) · Zbl 0823.35159 |
| [84] | Lange, H.; Zweifel, P., Quantum BGK models for the Wiegner-Poisson equation, Transp. Theory Statist. Phys., 25, 713-722 (1996) · Zbl 0869.35087 |
| [85] | Zweifel, P., The Wiegner transform and the Wiegner-Poisson system, Transp. Theory Statist. Phys., 22, 459-484 (1993) · Zbl 0785.35102 |
| [86] | Zweifel, P., The Wiegner-Poisson system: A quantum transport equation, Reports Math. Phys., 33, 317-323 (1993) · Zbl 0811.35114 |
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