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Dynamical scaling in Smoluchowski's coagulation equations: uniform convergence
Govind Menon and Robert L. Pego
We consider the approach to self-similarity (or dynamical scaling) in Smoluchowski's coagulation equations for the solvable kernels $K(x,y)=2,x+y$ and $xy$.
We prove the uniform convergence of densities to the self-similar solution with exponential tails under the regularity hypothesis that a suitable moment have an integrable Fourier transform. For the discrete equations 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 are simple and rely on the Fourier inversion formula and the solution by the method of characteristics for the Laplace transform.