Skip to main content
SpringerLink
Log in

On coalescence equations and related models

  • Chapter
Modeling and Computational Methods for Kinetic Equations

Abstract

The Smoluchowski coalescence equation is a mean-field model for the growth of particles by successive mergers, and has been recently studied by deterministic and probabilis- tic methods. The present review article focuses on the deterministic approach and attempts to survey the currently available results on the questions of existence, uniqueness, mass conserva- tion, gelation, and large time behaviour, while sketching the needed mathematical tools. When fragmentation is also taken into account and a detailed balance condition is assumed, recent techniques used to investigate the trend to equilibrium are outlined.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Aizenman, M., Bak, T.A.: Convergence to equilibrium in a system of reacting polymers. Comm. Math. Phys., 65,203-230 (1979).

    Article  MathSciNet  MATH  Google Scholar 

  2. Aldous, D.J.: Deterministic and stochastic models for coalescence (aggregation, coagu- lation): a review of the mean-field theory for probabilists. Bernoulli, 5, 3–48 (1999).

    Article  MathSciNet  MATH  Google Scholar 

  3. Amann, H.: Coagulation-fragmentation processes.Arch. RationalMech. Anal, 151,339– 366 (2000).

    Article  MathSciNet  MATH  Google Scholar 

  4. Ane\C, Blachere, S.,Chafai, D.,Fougeres, P.,Gentil, I.,Malrieu, F.,Roberto, C, Scheffer, G. Sur les Ine'galite's de Sobolev Logarithmiques. Panoramas et Syntheses, 10, Socidte' Mathe'matique de France, Paris (2000).

    Google Scholar 

  5. Ball, J.M., Carr, J.: Asymptotic behaviour of solutions to the Becker-Doring equations for arbitrary initial data. Proc. Roy. Soc. Edinburgh Sect. A, 108,109–116 (1988).

    Article  MathSciNet  MATH  Google Scholar 

  6. Ball, J.M., Carr, J.: The discrete coagulation-fragmentation equations: existence, uniqueness, and density conservation. J. Statist. Phys., 61,203–234 (1990).

    Article  MathSciNet  MATH  Google Scholar 

  7. Ball, J.M., Carr, J., Penrose, O.: The Becker-Doring cluster equations : basic properties and asymptotic behaviour of solutions. Comm. Math. Phys., 104, 657–692 (1986).

    Article  MathSciNet  MATH  Google Scholar 

  8. Baranger, C: Collisions, coalescences et fragmentations des gouttelettes dans un spray: ecriture precise des equations relatives au modele TAB. Mdmoire de DEA, ENS de Cachan (2001).

    Google Scholar 

  9. Benilan, Ph., Wrzosek, D.: On an infinite system of reaction-diffusion equations. Adv. Math. Sci. Appl., 7, 349–364 (1997).

    MathSciNet  Google Scholar 

  10. Bertoin, J.: Eternal solutions to Smoluchowski's coagulation equation with additive kernel and their probabilistic interpretations. Ann. Appl. Prohab., 12, 547–564 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  11. Bobylev, A.V., Illner, R.: Collision integrals for attractive potentials.J. Statist. Phys., 95, 633–649 (1999).

    Article  MathSciNet  MATH  Google Scholar 

  12. Burobin, A. V.: Existence and uniqueness of a solution of a Cauchy problem for the inhomogeneous three-dimensional coagulation equation. Differential Equations, 19, 1187–1197(1983).

    MathSciNet  MATH  Google Scholar 

  13. 13. Canizo Ricon, J.A.: On the discrete coagulation-fragmentation equation, in preparation.

    Google Scholar 

  14. Carr, J.: Asymptotic behaviour of solutions to the coagulation-fragmentation equations. I. The strong fragmentation case. Proc. Roy. Soc. Edinburgh Sect. A, 111, 231–244 (1992).

    Article  MathSciNet  Google Scholar 

  15. Carr, J., da Costa, F.P.: Instantaneous gelation in coagulation dynamics. Z Angew. Math. Phys. 43, 974–983 (1992).

    Article  MathSciNet  MATH  Google Scholar 

  16. Carr, J., da Costa, F.P: Asymptotic behavior of solutions to the coagulation-fragmentation equations. II. Weak fragmentation. J. Statist. Phys., 11, 89–123 (1994).

    Article  Google Scholar 

  17. Collet, J.F., Goudon, T, Poupaud, F, Vasseur, A.: The Becker-Doring system and its Lifshitz-Slyozov limit. SI AM J. Appl. Math., 62, 1488–1500 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  18. Collet, J.F, Poupaud, F: Existence of solutions to coagulation-fragmentation systems with diffusion. Transport Theory Statist. Phys., 25, 503–513 (1996).

    Article  MathSciNet  MATH  Google Scholar 

  19. Collet, J.F, Poupaud, F: Asymptotic behaviour of solutions to the diffusive fragmentation-coagulation system. Phys. D, 114, 123–146 (1998).

    Article  MathSciNet  MATH  Google Scholar 

  20. da Costa, F.P.: Existence and uniqueness of density conserving solutions to the coagulation-fragmentation equations with strong fragmentation. J. Math. Anal. Appl, 192,892–914(1995).

    Article  MathSciNet  MATH  Google Scholar 

  21. da Costa, FP: On the dynamic scaling behaviour of solutions to the discrete Smolu-chowski equations. Proc. Edinburgh Math. Soc. (2), 39, 547–559 (1996)

    MATH  Google Scholar 

  22. da Costa, F.P: Asymptotic behaviour of low density solutions to the generalized Becker-Doring equations. NoDEA Nonlinear Differential Equations Appl, 5, 23–37 (1998).

    Article  MathSciNet  MATH  Google Scholar 

  23. da Costa, F.P.: A finite-dimensional dynamical model for gelation in coagulation processes. J. Nonlinear Sci., 8, 619–653 (1998).

    Article  MathSciNet  MATH  Google Scholar 

  24. da Costa, F.R: Convergence to equilibna of solutions to the coagulation-fragmentation equations. In: Nonlinear evolution equations and their applications (Macau, 1998), 45–56, World Sci. Publishing, River Edge, NJ (1999).

    Google Scholar 

  25. Cueille, S., Sire, C: Nontrivial polydispersity exponents in aggregation models. Phys. Rev. E, 55, 5465–5478 (1997).

    Article  Google Scholar 

  26. Deaconu, M., Fournier, N.: Probabilistic approach of some discrete and continuous coagulation equations with diffusion. Stochastic Process. Appl, 101, 83–111 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  27. Deaconu, M., Fournier, N., Tanre E.: A pure jump Markov process associated with Smoluchowski's coagulation equation. Ann. Probab., 30, 1763–1796 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  28. Deaconu, M., Tanrd, E.: Smoluchowski's coagulation equation: probabilistic interpretation of solutions for constant, additive and multiplicative kernels. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 29, 549–579 (2000).

    MathSciNet  MATH  Google Scholar 

  29. van Dongen, P.G.J.: On the possible occurrence of instantaneous gelation in Smolu-chowski’s coagulation equation. J. Phys. A, 20, 1889–1904 (1987).

    Article  Google Scholar 

  30. van Dongen, P.G.J., Ernst, M.H.: Cluster size distribution in irreversible aggregation at large times. J. Phys. A, 18, 2779–2793 (1985).

    Article  MathSciNet  Google Scholar 

  31. van Dongen, P.G.J., Ernst, M.H.: On the occurrence of a gelation transition in Smolu-chowski's coagulation equation. J. Statist. Phys., 44, 785–792 (1986).

    Article  MathSciNet  Google Scholar 

  32. van Dongen, P.G.J., Ernst, M.H.: Scaling solutions of Smoluchowski's coagulation equation. Statist. Phys., 50, 295–329 (1988).

    Article  MathSciNet  MATH  Google Scholar 

  33. Drake, R.L.: A general mathematical survey of the coagulation equation. In: Topics in Current Aerosol Research (part 2), International Reviews in Aerosol Physics and Chemistry, 203–376, Pergamon Press, Oxford (1972).

    Google Scholar 

  34. Dubovskii, P.B.: Mathematical Theory of Coagulation. Lecture Notes Sen 23 Seoul Nat. Univ., Seoul (1994).

    Google Scholar 

  35. Dubovskii, P.B., Stewart, I.W.: Existence, uniqueness and mass conservation for the coagulation-fragmentation equation. Math. Methods Appl. Sci., 19, 571–591 (1996).

    Article  MathSciNet  MATH  Google Scholar 

  36. Durrett, R., Granovsky, B.L., Gueron, S.: The equilibrium behavior of reversible coagulation-fragmentation processes. J. Theoret. Probab., 12,447–474 (1999).

    Article  MathSciNet  MATH  Google Scholar 

  37. Eibeck, A., Wagner, W: Approximative solution of the coagulation-fragmentation equation by stochastic particle systems. Stochastic Anal. Appl, 18, 921–948 (2000).

    Article  MathSciNet  MATH  Google Scholar 

  38. Ernst, M.H., Ziff, R.M., Hendriks, E.M.: Coagulation processes with a phase transition. J. Colloid Interface Sci., 97, 266–277 (1984).

    Article  Google Scholar 

  39. Escobedo, M., Laurençot, Ph., Mischler, S.: Fast reaction limit of the discrete diffusive coagulation-fragmentation equation. Comm. Partial Differential Equations, 28, 1113–1133(2003).

    Article  MathSciNet  MATH  Google Scholar 

  40. 40. Escobedo, M., Laurencçot, Ph., Mischler, S.: On a kinetic equation for coalescing particles. Comm. Math. Phys., to appear.

    Google Scholar 

  41. Escobedo, M., Laurençjot, Ph., Mischler, S., Perthame, B.: Gelation and mass conservation in coagulation-fragmentation models. J. Differential Equations, 195, 143–174 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  42. Escobedo, M., Mischler, S., Perthame, B.: Gelation in coagulation and fragmentation models. Comm. Math. Phys., 231, 157–188 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  43. Fasano, A., Rosso, F.: A new model for the dynamics of dispersions in a batch reactor. In: Lectures on Applied Mathematics, A.J. Burgartz, R.H.W. Hoppe, C. Zenger (eds.), Springer-Verlag, New York, 2000,123–142.

    Google Scholar 

  44. 44. Filbet, F, Laurençot, Ph.: Numerical simulation of the Smoluchowski coagulation equation. SI AM J. Sci. Comput., to appear.

    Google Scholar 

  45. 45. Fournier, N., Mischler, S.: A Boltzmann equation for elastic, inelastic and coalescing collisions, submitted.

    Google Scholar 

  46. Friedlander, S.K., Wang, C.S.: The self-preserving particle size distribution for coagulation by brownian motion. J. Colloid Interface Sci., 22,126–132 (1966).

    Article  Google Scholar 

  47. Galkin, V.A., Dubovskii, P.B.: Solution of the coagulation equations with unbounded kernels. Differential Equations, 22, 373–378 (1986).

    MathSciNet  MATH  Google Scholar 

  48. Golovin, A.M.: The solution of the coagulation equation for cloud droplets in a rising air current. Izv. Geophys. Sen, 5,482–487 (1963).

    Google Scholar 

  49. Guia§, F.: Convergence properties of a stochastic model for coagulation-fragmentation processes with diffusion. Stochastic Anal. Appl., 19, 245–278 (2001).

    Article  MathSciNet  Google Scholar 

  50. Hendriks, E.M., Ernst, M.H., Ziff, R.M.: Coagulation equations with gelation. J. Statist. Phys. 31, 519–563 (1983).

    Article  MathSciNet  Google Scholar 

  51. Herrero, M.A., Velázquez, J.J.L., Wrzosek, D.: Solgel transition in a coagulation-diffusion model. Phys. D, 141, 221–247 (2000).

    Article  MathSciNet  MATH  Google Scholar 

  52. Jabin, P.-E.: Various levels of models for aerosols. Math. Models Methods Appl Sci, 12, 903–919 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  53. Jabin, P.-E., Niethammer, B.: On the rate of convergence to equilibrium in the Becker-Döring equations. J. Differential Equations, 191, 518–543 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  54. Jabin, P.-E., Perthame, B.: Notes on mathematical problems on the dynamics of dispersed particles interacting through a fluid. In: Modeling in applied sciences, 111–147, Model. Simul. Sci. Eng. Technol., Birkhauser Boston, Cambridge, MA, 2000.

    Google Scholar 

  55. Jeon, I.: Existence of gelling solutions for coagulation-fragmentation equations. Comm. Math. Phys., 194, 541–567 (1998).

    Article  MathSciNet  MATH  Google Scholar 

  56. Jourdain, B.: Nonlinear processes associated with the discrete Smoluchowski coagulation fragmentation equation. Markov Process. Related Fields, 9, 103–130 (2003).

    MathSciNet  MATH  Google Scholar 

  57. Kokholm, N.J.: On Smoluchowski’s coagulation equation. J. Phys. A, 21,839–842 (1988).

    Article  MathSciNet  MATH  Google Scholar 

  58. Kreer, M., Penrose, O.: Proof of dynamical scaling in Smoluchowski’s coagulation equation with constant kernel. J. Statist. Phys., 75, 389–407 (1994).

    Article  MathSciNet  MATH  Google Scholar 

  59. Krivitsky, D.S.: Numerical solution of the Smoluchowski kinetic equation and asymptotics of the distribution function. J. Phys. A, 28, 2025–2039 (1995).

    Article  MathSciNet  MATH  Google Scholar 

  60. Lang, R., Nguyen Xuan Xanh: Smoluchowski’s theory of coagulation in colloids holds rigorously in the Boltzmann-Grad-limit. Z. Wahrscheinlichkeitstheorie verw. Gebiete, 54, 227–280(1980).

    Google Scholar 

  61. Laurençot, Ph.: On a class of continuous coagulation-fragmentation equations. J. Differential Equations, 167, 145–174 (2000).

    Article  Google Scholar 

  62. Laurençot, Ph.: The discrete coagulation equation with multiple fragmentation. Proc. Edinburgh Math. Soc. (2), 45, 67–82 (2002).

    MATH  Google Scholar 

  63. Laurençot, Ph., Mischler, S.: Global existence for the discrete diffusive coagulation-fragmentation equation in L. Rev. Mat. Iheroamericana, 18, 731–745 (2002).

    Article  MATH  Google Scholar 

  64. Laurençot, Ph., Mischler, S.: The continuous coagulation-fragmentation equations with diffusion. Arch. RationalMech. Anal, 162, 45–99 (2002).

    Article  MATH  Google Scholar 

  65. Laurençot, Ph., Mischler, S.: From the discrete to the continuous coagulation-fragmentation equations. Proc. Roy. Soc. Edinburgh Sect. A, 132, 1219–1248 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  66. Laurençot, Ph., Mischler, S.: From the Becker-Döring to the Lifshitz-Slyozov-Wagner equations. J. Statist. Phys., 106, 957–991 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  67. Laurençot, Ph., Mischler, S.: Convergence to equilibrium for the continuous coagulation-fragmentation equation. Bull. Sci Math., 127, 179–190 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  68. Laurençot, Ph., Mischler, S.: Une introduction aux équations de coagulation-fragmentation. Book in preparation.

    Google Scholar 

  69. Laurençot, Ph., Wrzosek, D.: The Becker-Döring model with diffusion. II. Long time behaviour. J. Differential Equations, 148, 268–291 (1998).

    Article  MathSciNet  MATH  Google Scholar 

  70. Laurençot, Ph., Wrzosek, D.: The discrete coagulation equation with colhsional breakage. J. Statist Phys., 104,193–220 (2001).

    Article  MathSciNet  MATH  Google Scholar 

  71. Lee, M.H.: A survey of numerical solutions to the coagulation equation. J. Phys. A, 34, 10219–10241 (2002).

    Article  Google Scholar 

  72. Leyvraz, F: Existence and properties of post-gel solutions for the kinetic equations of coagulation. J. Phys. A, 16, 2861–2873 (1983).

    Article  MathSciNet  Google Scholar 

  73. Leyvraz, F, Tschudi, H.R.: Singularities in the kinetics of coagulation processes. J. Phys. A, 14, 3389–3405 (1981).

    Article  MathSciNet  MATH  Google Scholar 

  74. Leyvraz, F, Tschudi, H.R.: Critical kinetics near gelation. J. Phys. A, 15, 1951–1964 (1982).

    Article  MathSciNet  Google Scholar 

  75. Lifshitz, I.M., Slyozov, V.V.: The kinetics of precipitation from supersaturated solid solutions. J. Phys. Chem. Solids, 19, 35–50 (1961).

    Article  Google Scholar 

  76. Lions, P.-L.: Compactness in Boltzmann’s equation via Fourier integral operators and applications I. J. Math. Kyoto Univ., 34, 391–427 (1994).

    MathSciNet  MATH  Google Scholar 

  77. Lushnikov, A.A.: Coagulation in finite systems. J. Colloid Interface Sci., 65, 276–285 (1978).

    Article  Google Scholar 

  78. McGrady, E.D., Ziff, R.M.: “Shattering” transition in fragmentation. Phys. Rev. Lett., 58, 892–895 (1987).

    Article  MathSciNet  Google Scholar 

  79. McLeod, J.B.: On an infinite set of non-linear differential equations. Quart. J. Math. Oxford Ser. (2), 13, 119–128 (1962).

    Article  MathSciNet  MATH  Google Scholar 

  80. McLeod, J.B.: On an infinite set of non-linear differential equations (II). Quart. J. Math. Oxford Ser. (2), 13,193–205 (1962).

    Article  MathSciNet  Google Scholar 

  81. McLeod, J.B.: On the scalar transport equation. Proc. London Math. Soc. (3), 14,445–458 (1964).

    Article  MathSciNet  MATH  Google Scholar 

  82. Marcus, A.H.: Stochastic coalescence. Technometrics, 10, 133–143 (1968).

    Article  MathSciNet  Google Scholar 

  83. Meesters, A., Ernst, M.H.: Numerical evaluation of self-preserving spectra in Smolu-chowski’s coagulation theory. J. Colloid Interface Sci., 119, 576–587 (1987).

    Article  Google Scholar 

  84. Melzak, Z.A.: A scalar transport equation. Trans. Amer. Math. Soc, 85, 547–560 (1957).

    Article  MathSciNet  MATH  Google Scholar 

  85. Menon, G., Pego, R.L.: Dynamical scaling in Smoluchowski’s coagulation equations: uniform convergence. Comm. Pure Appl. Math., to appear.

    Google Scholar 

  86. Menon, G., Pego, R.L.: Approach to self-similarity in Smoluchowski’s coagulation equa-tions. Comm. Pure Appl. Math., to appear.

    Google Scholar 

  87. Mischler, S., Rodriguez Ricard, M.: Existence globale pour 1’équation de Smoluchowski continue non homogène et comportement asymptotique des solutions. C. R. Acad. Sci. Paris Sér. I Math., 336,407–412 (2003).

    Article  MATH  Google Scholar 

  88. Mischler, S., Wennberg, B.: On the spatially homogeneous Boltzmann equation. Ann. Inst. H. Poincare Anal. Non Lin éaire, 16, 467–501 (1999).

    Article  MathSciNet  MATH  Google Scholar 

  89. Müller, H.: Zur allgemeinen Theorie der raschen Koagulation. Kolloidchemische Beihefte, 27,223–250 (1928).

    Article  Google Scholar 

  90. Niethammer, B.: On the evolution of large clusters in the Becker-Döring model. J. Non-linear Science, 13,115–155 (2003).

    MathSciNet  MATH  Google Scholar 

  91. Norris, J.R.: Smoluchowski’s coagulation equation : uniqueness, non-uniqueness and a hydrodynamic limit for the stochastic coalescent. Ann. Appl. Probab., 9,78–109 (1999).

    Article  MathSciNet  MATH  Google Scholar 

  92. Norris, J.R.: Cluster coagulation. Comm. Math. Phys., 209, 407–435 (2000).

    Article  MathSciNet  MATH  Google Scholar 

  93. Penrose, O.: The Becker-Döring equations at large times and their connection with the LSW theory of coarsening. J. Statist. Phys., 89, 305–320 (1997).

    Article  MathSciNet  MATH  Google Scholar 

  94. Penrose, O., Lebowitz, J.L., Marro, J., Kalos, M.H., Sur, A.: Growth of clusters in a first-order phase transition. J. Statist. Phys., 19, 243–267 (1978).

    Article  Google Scholar 

  95. Roquejoffre, J.M., Villedieu, Ph.: A kinetic model for droplet coalescence in dense sprays. Math. Models Methods Appl. Sci., 11, 867–882 (2001).

    Article  MathSciNet  MATH  Google Scholar 

  96. Slemrod, M.: Trend to equilibrium in the Becker-Döring cluster equations. Nonlinearity, 2,429–443 (1989).

    Article  MathSciNet  MATH  Google Scholar 

  97. Slemrod, M.: A note on the kinetic equations of coagulation. J. Integral Equations Appl., 3,167–173 (1991).

    Article  MathSciNet  MATH  Google Scholar 

  98. Slemrod, M.: The Becker-Döring equations. In: Modeling in Applied Sciences, 149–171, Model. Simul. Sci. Eng. Technol., Birkhäuser Boston, Cambridge, MA (2000).

    Google Scholar 

  99. Smoluchowski, M.: Drei Vorträge über Diffusion, Brownsche Molekularbewegung und Koagulation von Kolloidteilchen. Physik. Zeitschr., 17, 557–599 (1916).

    Google Scholar 

  100. Smoluchowski, M.: Versuch einer mathematischen Theorie der Koagulationskinetik kol-loider Lösungen. Zeitschrift f. physik. Chemie, 92,129–168 (1917).

    Google Scholar 

  101. Spouge, J.L.: An existence theorem for the discrete coagulation-fragmentation equations. Math. Proc. Cambridge Philos. Soc., 96, 351–357 (1984).

    Article  MathSciNet  MATH  Google Scholar 

  102. Stewart, I. W.: A global existence theorem for the general coagulation-fragmentation equa-tion with unbounded kernels. Math. Methods Appl. Sci., 11, 627–648 (1989).

    Article  MathSciNet  MATH  Google Scholar 

  103. Stewart, I.W.: On the coagulation-fragmentation equation. Z. Angew. Math. Phys., 41, 917–924(1990).

    Article  MathSciNet  MATH  Google Scholar 

  104. Stewart, I.W.: A uniqueness theorem for the coagulation-fragmentation equation. Math. Proc. Cambridge Philos. Soc, 107, 573–578 (1990).

    Article  MathSciNet  MATH  Google Scholar 

  105. Stewart, I.W.: Density conservation for a coagulation equation. Z Angew. Math. Phys., 42, 746–756(1991).

    Article  MathSciNet  MATH  Google Scholar 

  106. Stewart, I.W., Dubovskiĭ, P.B.: Approach to equilibrium for the coagulation-fragmentation equation via a Lyapunov functional. Math. Methods Appl. Sci., 19, 171–185 (1996).

    Article  MathSciNet  MATH  Google Scholar 

  107. Stockmayer, W.H.: Theory of molecular size distribution and gel formation in branched-chain polymers. J. Chem. Phys., 11, 45–55 (1943).

    Article  Google Scholar 

  108. Villedieu, Ph., Hylkema, J.: Une méthode particulaire aléatoire reposant sur une équation cinétique pour la simulation numérique des sprays denses de gouttelettes liquides. C. R. Acad. Sci. Paris Sér I Math., 325, 323–328 (1997).

    Article  MathSciNet  MATH  Google Scholar 

  109. Wagner, C: Theorie der Alterung von Niederschlägen durch Umlosen (Ostwald-Reifung). Z. Elektrochem., 65, 581–591 (1961).

    Google Scholar 

  110. Wagner, W: Stochastic, analytic and numerical aspects of coagulation processes. Math. Comp. Simul, 62, 265–275 (2003).

    Article  MATH  Google Scholar 

  111. White, W.H.: A global existence theorem for Smoluchowski’s coagulation equations. Proc. Amer. Math. Soc, 80, 273–276 (1980).

    MathSciNet  MATH  Google Scholar 

  112. Wilkins, D.: A geometrical interpretation of the coagulation equation. J. Phys. A, 15, 1175–1178(1982).

    MathSciNet  Google Scholar 

  113. Wrzosek, D.: Existence of solutions for the discrete coagulation-fragmentation model with diffusion. Topol. Methods Nonlinear Anal., 9, 279–296 (1997).

    MathSciNet  MATH  Google Scholar 

  114. Wrzosek, D.: Mass-conserving solutions to the discrete coagulation-fragmentation model with diffusion. Nonlinear Anal, 49, 297–314 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  115. Ziff, R.M.: Kinetics of polymerization. J. Statist. Phys., 23, 241–263 (1980).

    Article  MathSciNet  Google Scholar 

  116. Ziff, R.M., Stell, G.: Kinetics of polymer gelation. J. Chem. Phys.,13,3492–3499 (1980).

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathématiques pour l’lndustrie et la Physique, CNRS UMR 5640, Université Paul Sabatier-Toulouse 3, 118 route de Narbonne, F-31062, Toulouse cedex 4, France

    Philippe Laurençot

  2. Ceremade - UMR 7534, Université de Paris IX - Dauphine, Place du Maréchal De Lattre de Tassigny, 75775, Paris Cedex 16, France

    Stéphane Mischler

Authors
  1. Philippe Laurençot
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Stéphane Mischler
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Mathematics, Université Paul Sabatier, 31 062, Toulouse Cedex, France

    Pierre Degond

  2. Department of Mathematics, Università di Ferrara, I-44100, Ferrara, Italy

    Lorenzo Pareschi

  3. Università di Catania, 95125, Catania, Italy

    Giovanni Russo

Rights and permissions

Reprints and permissions

Copyright information

© 2004 Springer Science+Business Media New York

About this chapter

Cite this chapter

Laurençot, P., Mischler, S. (2004). On coalescence equations and related models. In: Degond, P., Pareschi, L., Russo, G. (eds) Modeling and Computational Methods for Kinetic Equations. Modeling and Simulation in Science, Engineering and Technology. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-0-8176-8200-2_11

Download citation

  • DOI: https://doi.org/10.1007/978-0-8176-8200-2_11

  • Publisher Name: Birkhäuser, Boston, MA

  • Print ISBN: 978-1-4612-6487-3

  • Online ISBN: 978-0-8176-8200-2

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation