Dynamical Scaling in Smoluchowski’s Coagulation Equations: Uniform Convergence

Smoluchowski’s coagulation equation is a fundamental mean-field model of clustering dynamics. We consider the approach to self-similarity (or dynamical scaling) of the cluster size distribution for the “solvable” rate kernels $K(x,y)=2,x+y$, and $xy$. In the case of continuous cluster size distributions, we prove uniform convergence of densities to a self-similar solution with exponential tail, under the regularity hypothesis that a suitable moment have an integrable Fourier transform. For discrete size distributions, we prove uniform convergence under optimal moment hypotheses. Our results are completely analogous to classical local convergence theorems for the normal law in probability theory. The proofs rely on the Fourier inversion formula and the solution for the Laplace transform by the method of characteristics in the complex plane.