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

Smoluchowski's coagulation equation for irreversible aggregation with constant kernel is considered in its discrete version[Figure not available: see fulltext.] where c_t=c₁(t) is the concentration of l-particle clusters at time t. We prove that for initial data satisfying c₁(0)>0 and the condition 0 =c_l(0) <A (1+?)^-l(A ?>0), the solutions behave asymptotically like c₁(t)~t^-2˜c(lt^-1) as t?8 with lt^-1 kept fixed. The scaling function ˜c(?) is (1/gr)?, where {Mathematical expression}, 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 {Mathematical expression} where c(v, t) is the oncentration of clusters of size v. © 1994 Plenum Publishing Corporation.