Proof of dynamical scaling in Smoluchowski's coagulation equation with constant kernel

Abstract

Smoluchowski's coagulation equation for irreversible aggregation with constant kernel is considered in its discrete version Open image in new window wherect=c1(t) is the concentration ofl-particle clusters at timet. We prove that for initial data satisfyingc1(0)>0 and the condition 0 ⩽cl(0) 0), the solutions behave asymptotically likec1(t)∼t−2≈c(lt−1) ast→∞ withlt−1 kept fixed. The scaling function ≈c(ξ) is (1/gr)ξ, where\(\rho = \sum _l lc_l (0)\), a conserved quantity, is the initial number of particles per unit volume. An analous result is obtained for the continuous version of Smoluchowski's coagulation equation\(\frac{\partial }{{\partial t}}c(v,{\text{ }}t) = \int_0^v {du{\text{ }}c(v - u,{\text{ }}t){\text{ }}c(u,{\text{ }}t) - 2c(v,{\text{ }}t)} \int_0^\infty {du{\text{ }}c(u,{\text{ }}t)}\) wherec(v, t) is the oncentration of clusters of sizev.