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**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.

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 |

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\textit{J. R. Norris}, Ann. Appl. Probab. 9, No. 1, 78--109 (1999; Zbl 0944.60082)

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