Regularity, local behavior and partial uniqueness for self-similar profiles of Smoluchowski’s coagulation equation. (English) Zbl 1242.82031

Cañizo, José A.; Mischler, Stéphane

Rev. Mat. Iberoam. 27, No. 3, 803-839 (2011).

The paper is concerned with the analysis of the Smoluchowski equation which models irreversible processes of aggregation of particles in clusters. The authors consider a continuum setting, where the size \(y\) of the cluster may be any positive number, and assume a homogeneous kernel depending on two parameters \(\alpha\) and \(\beta\), subjected to some limitations. It is proved that the self-similar solutions are infinitely differentiable and sharp results on the behaviour of the self-similar profiles at \(y = 0\) are given in the case \(\alpha < 0\). Moreover, partial uniqueness results are presented. In particular, when \(\alpha = 0\), it is proved that two profiles with same mass and same momentum of order \(\alpha + \beta\) are necessarily equal, while when \(\alpha < 0\), it is proved that two profiles with the same moments of order \(\alpha\) and \(\beta\) and with the same asymptotic behaviour at \(y=0\) are equal. The proofs are based on a distributional representation of the coagulation operator along with estimates of its regularity by using the theory of the derivatives of fractional order.

Reviewer: Vittorio Romano (Catania)

MSC:

82C21	Dynamic continuum models (systems of particles, etc.) in time-dependent statistical mechanics
45K05	Integro-partial differential equations
82C05	Classical dynamic and nonequilibrium statistical mechanics (general)

Keywords:

coagulation; self-similarity; regularity; uniqueness; asymptotic behavior

References:

[1]	Aldous, D. J.: Deterministic and stochastic models for coalescence (ag- gregation, coagulation): a review of the mean-field theory for probabilists. Bernouilli 5 (1999), no. 1, 3-48. · Zbl 0930.60096
[2]	Cueille, S. and Sire, C.: Nontrivial polydispersity exponents in aggre- gation models. Phys. Rev. E 55 (1997), 5465-5478.
[3]	Drake, R. L.: A general mathematical survey of the coagulation equa- tion. In Topics in Current Aerosol Research (Part 2), volume 3, 201-376. International Reviews in Aerosol Physics and Chemistry. Pergamon, 1972.
[4]	Escobedo, M. and Mischler, S.: Dust and self-similarity for the Smolu- chowski coagulation equation. Ann. Inst. H. Poincaré Anal. Non Linéaire 23 (2006), no. 3, 331-362. · Zbl 1154.82024
[5]	Escobedo, M., Mischler, S. and Ricard, R. M.: On self-similarity and stationary problem for fragmentation and coagulation models. Ann. Inst. H. Poincaré Anal. Non Linéaire 22 (2005), no. 1, 99-125. · Zbl 1130.35025
[6]	Fournier, N. and Laurenc \?ot, P.: Existence of self-similar solutions to Smoluchowski’s coagulation equation. Comm. Math. Phys. 256 (2005), no. 3, 589-609. · Zbl 1084.82006
[7]	Fournier, N. and Laurenc \?ot, P.: Local properties of self-similar solu- tions to Smoluchowski’s coagulation equation with sum kernels. Proc. Roy. Soc. Edinburgh Sect. A 136 (2006), no. 3, 485-508. · Zbl 1229.45013
[8]	Hartman, P.: Ordinary Differential Equations. Classics in Applied Math- ematics 38. Society for Industrial and Applied Mathematics, Philadelphia, PA, 2002. · Zbl 1009.34001
[9]	Kreer, M. and Penrose, O.: Proof of dynamical scaling in Smolu- chowski’s coagulation equation with constant kernel. J. Statist. Phys. 75 (1994), no. 3-4, 389-407. · Zbl 0828.60093
[10]	Laurenc \?ot, P. and Mischler, S.: On coalescence equations and related models. In Modeling and computational methods for kinetic equations, 321-356. Model. Simul. Sci. Eng. Technol. Birkhäuser, Boston, MA, 2004. · Zbl 1105.82027
[11]	Leyvraz, F.: Scaling theory and exactly solved models in the kinetics of irreversible aggregation. Physics Reports 383 (2003), no. 2-3, 95-212. 839
[12]	Menon, G. and Pego, R. L.: Approach to self-similarity in Smolu- chowski’s coagulation equations. Comm. Pure Appl. Math. 57 (2004), no. 9, 1197-1232. · Zbl 1049.35048
[13]	Menon, G. and Pego, R. L.: Dynamical scaling in Smoluchowski’s coag- ulation equations: uniform convergence. SIAM J. Math. Anal. 36 (2005), no. 5, 1629-1651. · Zbl 1130.35128
[14]	Oldham, K. B. and Spanier, J.: The fractional calculus; theory and ap- plications of differentiation and integration to arbitrary order. Mathematics in Science and Engineering, V. Academic Press, 1974. · Zbl 0292.26011
[15]	Samko, S. G., Kilbas, A. A. and Marichev, O. I.: Fractional integrals and derivatives: Theory and applications. CRC, 1993. · Zbl 0818.26003
[16]	Schwartz, L.: Théorie des distributions. Hermann, 1997.
[17]	Van Dongen, P. G. J. and Ernst, M. H.: Scaling solutions of Smolu- chowski’s coagulation equation. J. Statist. Phys. 50 (1988), no. 1-2, 295-329. · Zbl 0998.65139