## Smoluchowski’s coagulation equation: Uniqueness, nonuniqueness and a hydrodynamic limit for the stochastic coalescent.(English)Zbl 0944.60082

Consider the mathematical model of the process of coagulation, which was proposed by Smoluchowski in 1916. Smoluchowski argued that particles of radius $$r$$ would perform independent Brownian motions of variance proportional to $$1/r$$, so pairs of particles of radii $$r_1$$ and $$r_2$$ would need at a rate proportional to $$(r_1+ r_2)(1/r_1+ 1/r_2)$$. The coagulation kernel is $$K(x,y)= (x^{1/3}+ y^{1/3})(x^{-1/3}+ y^{-1/3})$$ for particles of masses $$x$$ and $$y$$. Smoluchowski wrote down the infinite system of differential equations for the evolution of densities $$\mu(x)$$ of particles of mass $$x= 1,2,3,\dots$$ ${d\over dt} \mu_t(x)= {1\over 2} \sum^{x- 1}_{y= 1} K(y, x-y)\mu_t(y)\mu_t(x- y)- \mu_t(x) \sum^\infty_{y= 1} K(x,y)\mu_t(y).$ In this paper, sufficient conditions are given for existence and uniqueness in Smoluchowski’s coagulation equations. In particular, the following are given:
1. The existence of solutions for continuous coagulation kernels $$K$$ such that $$K(x,y)/xy\to 0$$ as $$(x,y)\to \infty$$.
2. Local existence and uniqueness of solutions when $$K(x,y)\leq \varphi(x)\varphi(y)$$ for some continuous sublinear function $$\varphi: E\to(0,\infty)$$, provided that the initial mass distribution $$\mu_0$$ satisfies $$\int_{(0,\infty)} \varphi(x)^2\mu_0(dx)< \infty$$.
3. Using the argument in 2, treat the case where $$K(x,y)$$ blows up as $$x\to 0$$ or $$y\to 0$$, also prove uniqueness in some cases when the mass distribution has no second, or even first, moment.
4. Proof without any local regularity conditions on $$K$$.
5. Note that the initial mass distribution is not necessarily discrete and it does not have a density with respect to Lebesgue measure. The author constructs an example of a coagulation kernel $$K$$ and an initial mass distribution $$\mu_0$$, such that Smoluchowski’s equation has at least two distinct solutions, both of which are conservative, in the sense that $\int_{(0,\infty)} x\mu_t(dx)= \int_{(0,\infty)} x\mu_0(dx)< \infty\quad\text{for all }t.$ Finally, a stochastic system of coagulating particles is considered, where particles of masses $$x$$ and $$y$$ coagulate at a rate proportional to $$K(x,y)$$. The stochastic coalescent is shown to converge weakly to the solution of Smoluchowski’s equation.
Reviewer: T.I.Jeon (Taejon)

### MSC:

 60H35 Computational methods for stochastic equations (aspects of stochastic analysis) 82C21 Dynamic continuum models (systems of particles, etc.) in time-dependent statistical mechanics
Full Text:

### References:

 [1] Aldous, D. J. (1998). Deterministic and stochastic models for coalescence (aggregation, coagulation): a review of the mean-field theory for probabilists. Bernoulli. · Zbl 0930.60096 [2] Ball, J. M. and Carr, J. (1990). The discrete coagulation-fragmentation equations: existence, uniqueness, and density conservation. J. Statist. Phy s. 61 203-234. · Zbl 1217.82050 [3] Chandrasekhar, S. (1943). Stochastic problems in physics and astronomy. Rev. Modern Phy s. 15 1-89. · Zbl 0061.46403 [4] Clark, J. M. C. and Katsouros, V. (1999). Stably coalescent stochastic froths. Adv. Appl. Probab. · Zbl 0931.60083 [5] Dubovski i, P. B. and Stewart, I. W. (1996). Existence, uniqueness and mass conservation for the coagulation-fragmentation equation. Math. Methods Appl. Sci. 19 571-591. · Zbl 0852.45016 [6] Ethier, S. N. and Kurtz, T. K. (1986). Markov Processes: Characterization and Convergence. Wiley, New York. · Zbl 0592.60049 [7] Heilmann, O. J. (1992). Analy tical solutions of Smoluchowski’s coagulation equation. J. Phy s. A 25 3763-3771. · Zbl 0754.92023 [8] Jakubowski, A. (1986). On the Skorokhod topology. Ann. Inst. H. Poincaré Probab. Statist. 22 263-285. · Zbl 0609.60005 [9] Jeon, I. (1998). Existence of getting solutions for coagulation-fragmentation equations. Commun. Math. Phy s. · Zbl 0910.60083 [10] Lushnikov, A. A. (1978). Certain new aspects of the coagulation theory. Izv. Acad. Sci. USSR Atmospher. Ocean Phy s. 14 738-743. [11] Marcus, A. H. (1968). Stochastic coalescence. Technometrics 10 133-143. JSTOR: [12] McLeod, J. B. (1962). On an infinite set of nonlinear differential equations. Quart. J. Math. Oxford 13 119-128. · Zbl 0109.31501 [13] McLeod, J. B. (1964). On the scalar transport equation. Proc. London Math. Soc. 14 445- 458. · Zbl 0123.29901 [14] Pollard, D. (1984). Convergence of Stochastic Processes. Springer, New York. · Zbl 0544.60045 [15] van Smoluchowski, M. (1916). Drei Vorträge über Diffusion, Brownsche Bewegung und Koagulation von Kolloidteilchen. physik. Z. 17 557-585. [16] White, W. H. (1980). A global existence theorem for Smoluchowski’s coagulation equations. Proc. Amer. Math. Soc. 80 273-276. JSTOR: · Zbl 0442.34003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.