Modelling aggregation-fragmentation phenomena from kinetic to macroscopic scales

Mathematical and Computer Modelling

Volumn 52, Issue 5-6, 2010, Pages 802-813

  Bellouquid, A ^a   Bianca, C ^b  

^a CADI AYYAD UNIVERSITY   (Morocco)

^b POLITECNICO DI TORINO   (Italy)

Author keywords

Active particles; Functional subsystems; Hydrodynamic limit; Kinetic theory; Living systems

Indexed keywords

ACTIVE PARTICLES; FUNCTIONAL SUBSYSTEMS; HYDRODYNAMIC LIMIT; LARGE SYSTEM; LIVING SYSTEMS; MACROSCOPIC EQUATION; MACROSCOPIC SCALE; MATHEMATICAL STRUCTURE; MICROSCOPIC SCALE; NON-LINEARITY; NONLINEAR INTEGRO-DIFFERENTIAL EQUATIONS; PHYSICAL PHENOMENA; QUALITATIVE ANALYSIS; VARIABLE NUMBER;

CONTROL NONLINEARITIES; FLUID DYNAMICS; HYDRODYNAMICS; INITIAL VALUE PROBLEMS; INTEGRODIFFERENTIAL EQUATIONS; KINETIC THEORY; NONLINEAR EQUATIONS;

AGGLOMERATION;

EID: 77954176604     PISSN: 08957177     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.mcm.2010.05.010     Document Type: Article

References (56)

1. Bellomo N., Bianca C., Delitala M. Complexity analysis and mathematical tools towards the modelling of living systems. Phys. Life Rev. 2009, 6:144-175.
2. Bellomo N., Forni G. Complex multicellular systems and immune competition: new paradigms looking for a mathematical theory. Curr. Topics Develop. Biol. 2008, 81:485-502.
3. Ajmone Marsan G., Bellomo N., Egidi M. Towards a mathematical theory of complex socio-economical systems by functional subsystems representation. Kinet. Relat. Models 2008, 1:249-278.
4. Ajmone Marsan G. New paradigms towards the modelling of complex systems in behavioral economics. Math. Comput. Modelling 2009, 50:584-597.
5. Camerer C.F., Loewenstein G., Rabin M. Advances in Behavioral Economics 2003, Princeton University Press.
6. Bianca C. On the modelling of space dynamics in the kinetic theory for active particles. Math. Comput. Modelling 2010, 51:72-83.
7. Smoluchowski M. Versuch einer mathematischen theorie der koagulationskinetik kolloider lösungen. Z. Phys. Chem. 1917, 92:129-168.
8. Smoluchowski M. Drei vorträge über diffusion, brownsche molekularbewegung und koagulation von kolloidteilchen. Physik. Zeitschr. 1916, 17:557-599.
9. Drake R.L. A general mathematical survey of the coagulation equation. Topics in Current Aerosol Research, 3 Part 2 1972, 201-376. Pergamon Press, Oxford. G.M. Hidy, J.R. Brock (Eds.).
10. Becker R., Döring W. Kinetische behandlung der keimbildung in übersättigten dämpfen. Ann. Phys. 1935, 24:719-752.
11. Oort J.H., van the Hulst H.C. Gas and smoke in interstellar space. Bull. Astron. Inst. Netherland 1946, 10:187-210.
12. Lachowicz M., Laurençot P., Wrzosek D. On the Oort-Hulst-Safronov coagulation equation and its relation to the Smoluchowski equation. SIAM J. Math. Anal. 2003, 34:1399-1421.
13. Aldous D.J. Deterministic and stochastic models for coalescence (aggregation, coagulation): review of the mean-field theory for probabilists. Bernouilli 1999, 5:3-48.
14. Wattis J.A.D. An introduction to mathematical models of coagulation-fragmentation processes: a discrete deterministic mean-field approach. Physica D 2006, 222:1-20.
15. Escobedo M., Mischler S., Perthame B. Gelation in coagulation and fragmentation models. Comm. Math. Phys. 2002, 231:157-188.
16. Ball J.M., Carr J. The discrete coagulation-fragmentation equations: existence, uniqueness and density conservation. J. Stat. Phys. 1990, 61:203-234.
17. Collet J., Popoud F. Existence of solutions to coagulation-fragmentation systems with diffusion. Transport Theory Statist. Phys 1996, 25:503-513.
18. da Costa F.P. Existence and uniqueness of density conserving solutions to the coagulation-fragmentation equations with strong fragmentation. J. Math. Anal. Appl. 1995, 192:892-914.
19. Laurencot P., Wrzosek D. The Becker-Döring model with diffusion. I. basic properties of solutions. Colloq. Math. 1998, 75:245-269.
20. Slemrod M. Coagulation-diffusion systems: derivation and existence of solutions for the diffuse-interface structure equations. Physica D 1990, 46:351-366.
21. Stewart I.W. A global existence theorem for the general coagulation fragmentation equation with unbounded kernels. Math. Methods Appl. Sci. 1989, 11:627-648.
22. Stewart I.W., Dubovskii P.B. Approach to equilibrium for the coagulation fragmentation equation via a Lyapunov functional. Math. Methods Appl. Sci. 1996, 19:171-185.
23. White W.H. Global existence theorem for Smoluchowski's coagulation equation. Proc. Amer. Math. Soc. 1980, 80:273-276.
24. Wrzosek D. Existence of solutions for the discrete coagulation-fragmentation model with diffusion. Topol. Methods Nonlinear Anal. 1997, 9:279-296.
25. McLaughlin D.J., Lamb W., McBride A.C. An existence and uniqueness result for a coagulation and multiple-fragmentation equation. SIAM J. Math. Anal. 1997, 28:1173-1190.
26. McLaughlin D.J., Lamb W., McBride A.C. A semigroup approach to fragmentation models. SIAM J. Math. Anal. 1997, 28:1158-1172.
27. Lachowicz M., Wrzosek D. A nonlocal coagulation-fragmentation model, Dedicated to the memory of Wieslaw Szlenk. Appl. Math. (Warsaw) 2000, 27:45-66.
28. Lasheras J.C., Eastwood C., Martínez-Bazán C., Montañés J.L. A review of statistical models for the break-up of an immiscibile fluid immersed into a fully developed turbulent flow. Int. J. Multiph. Flow 2002, 28:242-278.
29. Fasano A., Rosso F., Mancini A. Implementation of a fragmentation-coagulation-scattering model for the dynamics of stirred liquid-liquid dispersions. Physica D 2006, 222:141-158.
30. Tabata M., Eshima N., Takagi I., Hiroyama T. The Cauchy problem for the system of equations describing migration motivated by regional economic disparity. Appl. Math. Comput. 1998, 94:45-64.
31. Othmer H.G., Stevens A. Aggregation, blowup, and collapse: the ABC's of taxis in reinforced random walks. SIAM J. Appl. Math. 1997, 57:1044-1081.
32. Zhao Z., Kirou A., Ruszczycki B., Johnson N.F. Dynamical clustering as a generator of complex system dynamics. Math. Models Methods Appl. Sci. 2009, 19(Suppl.):1539-1565.
33. Short M.B., D'orsogna M.R., Pasour V.B., Tita G.E., Brantingham P.J., Bertozzi A.L., Chayes L.B. A statistical model of criminal behavior. Math. Models Methods Appl. Sci. 2008, 18(Suppl.):1249-1267.
34. Bertotti M.L., Delitala M. Conservation laws and asymptotic behavior of a model of social dynamics. Nonlinear Anal. RWA 2008, 9:183-196.
35. Bertotti M.L., Delitala M. On the existence of limit cycles in opinion formation processes under time periodic influence of persuaders. Math. Models Methods Appl. Sci. 2008, 18:913-934.
36. Bellomo N., Delitala M. From the mathematical kinetic, and stochastic game theory to modelling mutations, onset, progression and immune competition of cancer cells. Phys. Life Rev. 2008, 5:183-206.
37. Szymamska Z., Rodrigo C.M., Lachowicz M., Chaplain M.A.J. Mathematical modelling of cancer invasion of tissue: the role and effect of nonlocal interactions. Math. Models Methods Appl. Sci. 2009, 19:257-283.
38. Bürger R., Garcia A., Kunik M. A generalized kinetic model of sedimentation of polydisperse suspensions with a continuous particle distribution. Math. Models Methods Appl. Sci. 2008, 18:1741-1786.
39. Jäger E., Segel Lee A. On the distribution of dominance in populations of social organisms. SIAM J. Appl. Math. 1992, 52:1442-1468.
40. Arlotti L., Bellomo N. Solution of a new class of nonlinear kinetic models of population dynamics. Appl. Math. Lett. 1996, 9:65-70.
41. Longo E., Bellomo N. The Boltzmann equation for dissipative mixtures. Appl. Math. Lett. 1999, 12:39-44.
42. Arlotti L., Bellomo N., Latrach K. From the Jäger and Segel model to kinetic population dynamics - nonlinear evolution problems and applications. Math. Comput. Modelling 1999, 30:15-40.
43. Bellomo N., Bellouquid A., Nieto J., Soler J. Multicellular growing systems: hyperbolic limits towards macroscopic description. Math. Models Methods Appl. Sci. 2007, 17:1675-1693.
44. N. Bellomo, A. Bellouquid, J. Nieto, J. Soler, Multiscale biological tissue models and flux-limited chemotaxis for multicellular growing systems, Math. Models Methods Appl. Sci. 20 (2010) (in press).
45. Bellomo N., Bellouquid A., Soler J. From the mathematical kinetic theory for active particles on the derivation of hyperbolic macroscopic tissue models. Math. Comput. Modelling 2009, 49:2083-2093.
46. Chalub F.A., Dolak-Struss Y., Markowich P., Oeltz D., Schmeiser C., Soref A. Model hierarchies for cell aggregation by chemotaxis. Math. Models Methods Appl. Sci. 2006, 16:1173-1198.
47. Arlotti L., Bellomo N., De Angelis E. Generalized kinetic (Boltzmann) models: mathematical structures and applications. Math. Models Methods Appl. Sci. 2002, 12:567-591.
48. Jabin P.-E., Lemesle V., Aurelle D. A continuous size-structured red coral growth model. Math. Models Methods Appl. Sci. 2009, 18:1927-1944.
49. De Lillo S., Delitala M., Salvatori C. Modelling epidemics and virus mutations by methods of the mathematical kinetic theory for active particles. Math. Models Methods Appl. Sci. 2009, 19:1405-1426.
50. Gallo I., Barra A., Contucci P. Parameter evaluation of a simple mean field mode of social interaction. Math. Models Methods Appl. Sci. 2009, 19:1427-1440.
51. C. Bianca, Mathematical modelling for keloid formation triggered by virus: malignant effects and immune system competition, Math. Models Methods Appl. Sci. 21 (2011) (in press).
52. Alesina A., Spolaore E. Conflict, defense spending and the number of nations. Eur. Econom. Rev. 2006, 50:91-120.
53. Aleskerov F., Kalyagin V., Pogorelskiy K. Actual voting power of the IMF members based on their political-economic integration. Math. Comput. Modelling 2008, 48:1554-1569.
54. Ansolabehere S., Leblanc W. A spatial model of the relationship between seats and votes. Math. Comput. Modelling 2008, 48:1409-1420.
55. Herrero M.A., Kön A., Perez-Pomares J.M. Modelling vascular morphogenesis: current views in blood vessels development. Math. Models Methods Appl. Sci. 2009, 19(Suppl.):1483-1538.
56. Bellomo N., Li N.K., Maini P.K. On the foundations of cancer modelling: selected topics, speculations, and perspectives. Math. Models Methods Appl. Sci. 2008, 18:593-646.