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

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Date
2010
Link to item file
http://hal.archives-ouvertes.fr/hal-00337661/en/
Dewey
Probabilités et mathématiques appliquées
Sujet
Smoluchowski's equation; coagulation equation; constant coagulation kernel; self-similar variables; spectral gap; exponential relaxation rate; explicit
Journal issue
SIAM Journal on Mathematical Analysis
Volume
41
Number
6
Publication date
2010
Article pages
2283-2314
Publisher
SIAM
DOI
http://dx.doi.org/10.1137/08074091X
URI
https://basepub.dauphine.fr/handle/123456789/3049
Collections
  • CEREMADE : Publications
Metadata
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Author
Cañizo, José Alfredo
Mischler, Stéphane
Mouhot, Clément
Type
Article accepté pour publication ou publié
Abstract (EN)
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.

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