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

TL;DR

This paper proves that solutions to Smoluchowski's coagulation equation with a constant kernel rapidly converge to a self-similar profile, with explicit rates and in strong Sobolev norms, using spectral analysis and transform methods.

Contribution

It establishes exponential convergence rates to self-similarity for the constant kernel case, employing spectral gap analysis and explicit transform solutions.

Findings

Exponential convergence in weighted Sobolev norms.

Spectral gap in the linearized coagulation operator.

Explicit solutions via Laplace/Fourier transforms.

Abstract

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.