Dynamical scaling in Smoluchowski's coagulation equations: Uniform convergence

SIAM Review

Volumn 48, Issue 4, 2006, Pages 745-768

  Menon, Govind ^a   Pego, Robert L ^b  

^a BROWN UNIVERSITY   (United States)

^b CARNEGIE MELLON UNIVERSITY   (United States)

Author keywords

Coarsening; Complex characteristics; Dynamic scaling; Self similar solutions

Indexed keywords

LAPLACE TRANSFORMS; MATHEMATICAL MODELS; OPTIMIZATION; PROBABILITY;

CLUSTER SIZE DISTRIBUTIONS; DYNAMIC SCALING; SMOLUCHOWSKI'S COAGULATION EQUATIONS; UNIFORM CONVERGENCE;

CONVERGENCE OF NUMERICAL METHODS;

EID: 33751427417     PISSN: 00361445     EISSN: None     Source Type: Journal    
DOI: 10.1137/060662496     Document Type: Article

References (27)

1. D. J. Aldous, Deterministic and stochastic models for coalescence (aggregation and coagulation): A review of the mean-field theory for probabilists, Bernoulli, 5 (1999), pp. 3-48.
2. J. Bertoin, Eternal solutions to Smoluchowski's coagulation equation with additive kernel and their probabilistic interpretations, Ann. Appl. Probab., 12 (2002), pp. 547-564.
3. F. P. Da Costa, On the dynamic scaling behaviour of solutions to the discrete Smoluchowski equations, Proc. Edinburgh Math. Soc. (2), 39 (1996), pp. 547-559.
4. R. L. Drake, A general mathematical survey of the coagulation equation, in Topics in Current Aerosol Research, G. M. Hidy and J. R. Brock, eds., International Reviews in Aerosol Physics and Chemistry, Pergamon, Oxford, 1972, pp. 201-376.
5. P. B. Dubovskiǐ and I. W. Stewart, Existence, uniqueness and mass conservation for the coagulation-fragmentation equation, Math. Methods Appl. Sci., 19 (1996), pp. 571-591.
6. M. Escobedo, S. Mischler, and B. Perthame, Gelation in coagulation and fragmentation models, Comm. Math. Phys., 231 (2002), pp. 157-188.
7. M. Escobedo, S. Mischler, and M. Rodriguez Ricard, On self-similarity and stationary problem for fragmentation and coagulation models, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), pp. 99-125.
8. W. Feller, An Introduction to Probability Theory and Its Applications, Vol. 2, 2nd ed., Wiley, New York, 1971.
9. N. Fournier and P. Laurençot, Existence of self-similar solutions to Smoluchowski's coagulation equation, Comm. Math. Phys., 256 (2005), pp. 589-609.
10. N. Fournier and P. Laurençot, Well-posedness of Smoluchowski coagulation equation for a class of homogeneous kernels, J. Funct. Anal., 233 (2006), pp. 351-379.
11. S. Friedlander, Smoke, Dust and Haze: Fundamentals of Aerosol Behavior, Wiley, New York, 1977.
12. I. Jeon, Existence of gelling solutions for coagulation-fragmentation equations, Comm. Math. Phys., 194 (1998), pp. 541-567.
13. Y. Katznelson, An Introduction to Harmonic Analysis, corrected ed., Dover, New York, 1976.
14. T. Kiorbøe, Formation and fate of marine snow: Small-scale processes with large-scale implications, Scientia Marina, 66 (2001), pp. 67-71.
15. M. Kreer and O. Penrose, Proof of dynamical scaling in Smoluchowski's coagulation equation with constant kernel, J. Statist. Phys., 75 (1994), pp. 389-407.
16. P. Laurençot and S. Mischler, Liapunov functionals for Smoluchowski's coagulation equation and convergence to self-similarity, Monatsh. Math., 146 (2005), pp. 127-142.
17. M. H. Lee, A survey of numerical solutions to the coagulation equation, J. Phys A, 34 (2001), pp. 10219-10241.
18. F. Leyvraz, Scaling theory and exactly solved models in the kinetics of irreversible aggregation, Phys. Rep., 383 (2003), pp. 95-212.
19. J. B. McLeod, On an infinite set of non-linear differential equations, Quart. J. Math. Oxford Ser. (2), 13 (1962), pp. 119-128.
20. G. Menon and R. Pego, Approach to self-similarity in Smoluchowski's coagulation equations, Comm. Pure Appl. Math., 57 (2004), pp. 1197-1232.
21. G. Menon and R. L. Pego, Ultimate dynamics and the scaling attractor for Smoluchowski's coagulation equations, in preparation.
22. H. S. Niwa, School size statistics offish, J. Theoret. Biol., 195 (1998), pp. 351-361.
23. J. R. Norris, Smoluchowski's coagulation equation: Uniqueness, nonuniqueness and a hydrodynamic limit for the stochastic coalescent, Ann. Appl. Probab., 9 (1999), pp. 78-109.
24. J. Silk and S. D. White, The development of structure in the expanding universe, Astrophys. J., 223 (1978), pp. L59-L62.
25. M. V. Smoluchowski, Drei Vorträge über Diffusion, Brownsche Bewegung und Koagulation von Kolloidteilchen, Physik. Z., 17 (1916), pp. 557-585.
26. P. G. J. van Dongen and M. H. Ernst, Scaling solutions of Smoluchowski's coagulation equation, J. Statist. Phys., 50 (1988), pp. 295-329.
27. R. M. Ziff, Kinetics of polymerization, J. Statist. Phys., 23 (1980), pp. 241-263.