Rate of Convergence to Self-Similarity for Smoluchowski's Coagulation Equation with Constant Coefficients

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 selfsimilar profile. This convergence holds in a weighted Sobolev norm which implies the L2 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.