Rate of convergence to self-similarity for Smoluchowski's coagulation equation with constant coefficients

Details Rate of convergence to self-similarity for Smoluchowski's coagulation equation with constant coefficients Rate of convergence to self-similarity for Smoluchowski's coagulation equation with constant coefficients

2008-11-07

arxiv.org/abs/0811.1169v1

SIAM Journal on Mathematical Analysis / SIAM Journal of Mathematical Analysis 41, 6 (2010) 2283-2314

Mathematics

Analysis of PDEs

41 pages

Scientific paper

10.1137/08074091X

We show that solutions to Smoluchowski's equation with a constant coagulation kernel and an initial datum with some regularity and exponentially decaying tail converge exponentially fast to a self-similar profile. This convergence holds in a weighted Sobolev norm which implies the L\^2 convergence of derivatives up to a certain order k depending on the regularity of the initial condition. We prove these results through the study of the linearized coagulation equation in self-similar variables, for which we show a spectral gap in a scale of weighted Sobolev spaces. We also take advantage of the fact that the Laplace or Fourier transforms of this equation can be explicitly solved in this case.

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