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

wherec _t =c ₁ (t) is the concentration ofl-particle clusters at timet. We prove that for initial data satisfyingc ₁(0)>0 and the condition 0 ⩽c _l (0) <A (1+Δ)^-l(A Δ>0), the solutions behave asymptotically likec ₁ (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.