PROOF OF DYNAMICAL SCALING IN SMOLUCHOWSKIS COAGULATION EQUATION WITHCONSTANT KERNEL

Smoluchowski's coagulation equation for irreversible aggregation with constant kernel is considered in its discrete version c(l) = SIGMA(k=1 )l-1 c(l-k)c(k) - 2c(l) SIGMA(k=1) infinity c(k) where c(l) = c(l)(t) is the concentration of l-particle clusters at time t. We prove that f or initial data satisfying c(l)(0) > 0 and the condition 0 less-than-o r-equal-to c(l)(0) < A(1 + DELTA)-l (A, DELTA > 0), the solutions beha ve asymptotically like c(l)(t) is similar to t-2c(lt-1) as t --> infin ity with lt-1 kept fixed. The scaling function c(xi) is (1/rho)exp[(-1 /rho)xi], where rho = SIGMA(l)lc(t)(0), a conserved quantity, is the i nitial number of particles per unit volume. An analogous result is obt ained for the continuous version of Smoluchowski's coagulation equatio n partial derivative/partial derivative t c(v,t) integral-v0 du c(v - u,t) c(u,t) - 2c(v,t) integral-infinity0 du c(u, t) where (c(v, t) is the concentration of clusters of size v.