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 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. © 2009 Society for Industrial and Applied Mathematics.
Original language | English |
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Pages (from-to) | 2283-2314 |
Journal | SIAM Journal on Mathematical Analysis |
Volume | 41 |
DOIs | |
Publication status | Published - 1 Jan 2009 |
Keywords
- Coagulation equation
- Constant coagulation kernel
- Explicit
- Self-similar variables
- Smoluchowski's equation
- Spectral gap