Existence of self-similar solutions to Smoluchowski’s coagulation equation with product kernel
Existence of self-similar solutions to Smoluchowski’s coagulation equation with product kernel
We explore, by using formal analysis, the existence of mass conserving self-similar solutions for Smoluchowski’s coagulation equation when kernel K(x, y) = x λ y µ + x µ y λ with 0 < λ + µ < 1.
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- [1] McLeod JB. On an infinite set of non-linear differential equations. The Quarterly Journal of Mathematics 1962; 13 (1): 119-128. doi: 10.1093/qmath/13.1.119
- [2] McLeod JB. On an infinite set of non-linear differential equations (II). The Quarterly Journal of Mathematics 1962; 13 (1): 193-205. doi: 10.1093/qmath/13.1.193
- [3] McLeod JB. On a recurrence formula in differential equations. The Quarterly Journal of Mathematics 1962; 13 (1): 283-284. doi: 10.1093/qmath/13.1.283
- [4] McLeod JB. On the scaler transport equation. Proceeding of the London Mathematical Society 1964; 14 (3): 445- 458. doi: 10.1112/plms/s3-14.3.445
- [5] McLeod JB, Niethammer B, Velázquez JJL. Asymptotics of self-similar solutions to coagulation equations with product kernel. Journal of Statistical Physics 2011; 144: 76-100. doi: 10.1007/s10955-011-0239-2
- [6] Hastings SP, McLeod JB. Methods in Ordinary Differential Equations: with Applications to Boundary Value Problems. Graduate Studies in Mathematics. Vol 129, Providence, USA: American Mathematical Society, 2012. doi: 10.1090/gsm/129
- [7] Menon G, Pego RL. Approach to self-similarity in Smoluchowski’s coagulation equations. Communications on Pure and Applied Mathematics 2004; 57 (9): 1197-1232. doi: 10.1002/cpa.3048
- [8] Ball JM, Carr J. The discrete coagulation-fragmentation equations: Existence, uniqueness, and density conservation. Journal of Statistical Physics 1990; 61 (1): 203-234. doi: 10.1007/BF01013961
- [9] Aldous DJ. Deterministic and stochastic models for coalescence (aggregation, coagulation): A review of the meanfield theory for probabilists. Bernoulli 1999; 5 (1): 3-48.
- [10] Melzak ZA. A scalar transport equation. Transactions of the American Mathematical Society 1957; 85: 547-560. doi: 10.1090/S0002-9947-1957-0087880-6
- [11] Escobedo M, Mischler S, Ricard MR. On self-similarity and stationary problem for fragmentation and coagulation models. Annales de L’Institut Henri Poincaré, Analyse Non Linéaire 2005; 22 (1): 99-125. doi: 10.1016/j.anihpc.2004.06.001
- [12] Smoluchowski M. Drei Vorträge über Diffusion, Brownsche Molekularbewegung und Koagulation von Kolloidteilchen. Physikalische Zeitschrift 1916; 17: 557-585 (in German).
- [13] Drake RL. A general mathematical survey of the coagulation equation. In: Hidy GM and Brock JR (Editors). Topics in Current Aerosol Research 3 (Part 2). Oxford, UK: Pergamon, 1972, pp. 201-376.
- [14] van Dongen PGJ, Ernst MHJ. Scaling solutions of Smoluchowski’s coagulation equation. Journal of Statistical Physics 1988; 50: 295-329. doi: 10.1007/BF01022996
- [15] Fournier N, Laurençot P. Existence of self-similar solutions to Smoluchowski’s coagulation equation. Communications in Mathematical Physics 2005; 256: 589-609. doi: 10.1007/s00220-004-1258-5
- [16] Granshteyn IS, Ryzhik IM. Table of Integrals, Series, and Products. Jeffrey A and Zwillinger D (Editors). Oxford, UK: Academic Press, 2007.