Dust and self-similarity for the Smoluchowski coagulation equation

Escobedo, M.; Mischler, S.

Annales de l'I.H.P. Analyse non linéaire, Tome 23 (2006) no. 3, pp. 331-362.

[1] Aldous D.J., Deterministic and stochastic models for coalescence (aggregation, coagulation): a review of the mean-field theory for probabilists, Bernoulli 5 (1999) 3-48. | MR | Zbl

[2] Bertoin J., The asymptotic behaviour of fragmentation processes, J. Eur. Math. Soc. (JEMS) 5 (4) (2003) 395-416. | MR | Zbl

[3] Bertoin J., Eternal solutions to Smoluchowski's coagulation equation with additive kernel and their probabilistic interpretations, Ann. Appl. Probab. 12 (2) (2002) 547-564. | MR | Zbl

[4] Bobylev A.V., Moment inequalities for the Boltzmann equation and applications to the spatially homogeneous problems, J. Statist. Phys. 88 (1997) 1183-1214. | MR | Zbl

[5] Bobylev A.V., Gamba I., Panferov V., Moment inequalities and high-energy tails for the Boltzmann equations with inelastic interactions, J. Statist. Phys. 116 (5-6) (2004) 1651-1682. | MR | Zbl

[6] Boccardo L., Gallouët T., Nonlinear elliptic equations with right-hand side measures, Comm. Partial Differential Equations 17 (3-4) (1992) 641-655. | MR | Zbl

[7] Cueille S., Sire C., Droplets nucleation and Smoluchovski's equation with growth and injection of particles, Phys. Rev. E 57 (1998) 881-900.

[8] Van Dongen P.G.J., Ernst M.H., Cluster size distribution in irreversible aggregation at large times, J. Phys. A 18 (1985) 2779-2793. | MR

[9] Van Dongen P.G.J., Ernst M.H., Solutions of Smoluchowski coagulation equation at large cluster sizes, Physica A 145 (1987) 15.

[10] Van Dongen P.G.J., Ernst M.H., Scaling solutions of Smoluchowski's coagulation equation, J. Statist. Phys. 50 (1988) 295-329. | MR | Zbl

[11] 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, Pergamon Press, Oxford, 1972, pp. 203-376.

[12] G. Duffa, N.T.-H. Nguyen-Bui, Un modèle de suies, Personal communication, 2002.

[13] Escobedo M., Mischler S., Perthame B., Gelation in coagulation and fragmentation models, Comm. Math. Phys. 231 (2002) 157-188. | MR | Zbl

[14] Escobedo M., Mischler S., Rodriguez Ricard M., On self-similarity and stationary problem for fragmentation and coagulation models, Ann. Inst. H. Poincaré Anal. Non Linéaire 22 (2005) 99-125. | EuDML | Numdam | MR | Zbl

[15] Fournier N., Giet J.-S., On small particles in coagulation-fragmentation equations, J. Statist. Phys. 111 (5-6) (2003) 1299-1329. | MR | Zbl

[16] N. Fournier, P. Laurençot, Existence of self-similar solutions to Smoluckovski's coagulation equation, Preprint, 2004. | MR | Zbl

[17] Friedlander S.K., Smoke, Dust and Haze: Fundamentals of Aerosol Behavior, Wiley, New York, 1979.

[18] Jeon I., Existence of gelling solutions for coagulation-fragmentation equations, Comm. Math. Phys. 194 (1998) 541-567. | MR | Zbl

[19] Kreer M., Penrose O., Proof of dynamical scaling in Smoluchowski's coagulation equation with constant kernel, J. Statist. Phys. 75 (1994) 389-407. | MR | Zbl

[20] Laurençot Ph., Mischler S., From the discrete to the continuous coagulation-fragmentation equations, Proc. Roy. Soc. Edinburgh Sect. A 132 (5) (2002) 1219-1248. | MR | Zbl

[21] Laurençot Ph., Mischler S., The continuous coagulation-fragmentation equations with diffusion, Arch. Rational Mech. Anal. 162 (2002) 45-99. | MR | Zbl

[22] Lê Châu-Hoàn, Etude de la classe des opérateurs m-accrétifs de $L^1\left(\Omega \right)$ et accrétifs dans $L^{\infty }\left(\Omega \right)$, Thèse de 3^ème cycle, Université de Paris VI, 1977.

[23] Laurençot P., Mischler S., On coalescence equations and related models, in: Degond P., Pareschi L., Russo G. (Eds.), Modelling and Computational Methods for Kinetic Equations, Modeling and Simulation in Science, Engineering and Technology (MSSET), Birkhäuser, 2004, pp. 321-356. | MR | Zbl

[24] P. Laurençot, S. Mischler, Coagulation and fragmentation equations, in preparation.

[25] Lee M.H., A survey on numerical solutions to the coagulation equation, J. Phys. A 34 (2001) 10219. | Zbl

[26] Leyvraz F., Existence and properties of post-gel solutions for the kinetic equations of coagulation, J. Phys. A 16 (1983) 2861-2873. | MR

[27] Leyvraz F., Scaling theory and exactly solved models in the kinetics of irreversible aggregation, Phys. Rep. 383 (2-3) (2003) 95-212.

[28] Lifshitz I.M., Slyozov V.V., The kinetics of precipitation from supersaturated solid solutions, J. Phys. Chem. Solids 19 (1961) 35-50.

[29] Lushnikov A.A., Kulmala M., Nucleation burst in coagulating system, Phys. Rev. E 62 (2000) 4932-4939.

[30] Mcgrady E.D., Ziff R.M., “Shattering” Transition in Fragmentation, Phys. Rev. Lett. 58 (1987) 892.

[31] Menon G., Pego R.L., Approach to self-similarity in Smoluchowski's coagulation equation, Comm. Pure Appl. Math. 57 (9) (2004) 1197-1232. | MR | Zbl

[32] G. Menon, R.L. Pego, Dynamical scaling in Smoluchowski's coagulation equation: uniform convergence, Preprint, 2003.

[33] Mischler S., Rodriguez Ricard M., Existence globale pour l'équation de Smoluchowski continue non homogène et comportement asymptotique des solutions, C. R. Acad. Sci. Paris, Ser. I Math. 336 (2003) 407-412. | MR | Zbl

[34] S. Mischler, Une introduction aux modèles de coagulation et fragmentation, Notes de cours de DEA, http://www.ceremade.dauphine.fr/~mischler/.

[35] Mischler S., Wennberg B., On the spatially homogeneous Boltzmann equation, Ann. Inst. H. Poincaré Anal. Non Linéaire 16 (4) (1999) 467-501. | EuDML | Numdam | MR | Zbl

[36] N. Morgan, C. Wells, M. Kraft, W. Wagner, Modelling nanoparticle dynamics: coagulation, sintering, particle inception and surface growth, Preprint No. 19, Cambridge Center for Computational Chemical Engineering, 2003.

[37] Norris J.R., Cluster coagulation, Comm. Math. Phys. 209 (2000) 407-435. | MR | Zbl

[38] Norris J.R., Smoluchowski's coagulation equation: uniqueness, non-uniqueness and a hydrodynamic limit for the stochastic coalescent, Ann. Appl. Probab. 9 (1999) 78-109. | MR | Zbl

[39] Seinfeld J.H., Atmospheric Chemistry and Physics of Air Pollution, John Wiley & Sons, New York, 1986.

[40] Smoluchowski M., Drei Vorträge über Diffusion, Brownsche Molekularbewegung und Koagulation von Kolloidteilchen, Physik Z. 17 (1916) 557-599.

[41] Smoluchowski M., Versuch einer mathematischen Theorie der Koagulationskinetik kolloider Lösungen, Z. Physik. Chemie 92 (1917) 129-168.

[42] Stewart I.W., A global existence theorem for the general coagulation-fragmentation equation with unbounded kernels, Math. Methods Appl. Sci. 11 (1989) 627-648. | MR | Zbl

[43] Stewart I.W., A uniqueness theorem for the coagulation-fragmentation equation, Math. Proc. Cambridge Philos. Soc. 107 (1990) 573-578. | MR | Zbl

[44] Pulvirenti A., Wennberg B., A Maxwellian lower bound for solutions to the Boltzmann equation, Comm. Math. Phys. 183 (1997) 145-160. | MR | Zbl