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

Cañizo, José Alfredo; Mischler, Stéphane; Mouhot, Clément

Cañizo, José Alfredo; Mischler, Stéphane; Mouhot, Clément (2010), Rate of convergence to self-similarity for Smoluchowski's coagulation equation with constant coefficients, SIAM Journal on Mathematical Analysis, 41, 6, p. 2283-2314. http://dx.doi.org/10.1137/08074091X

Type

Article accepté pour publication ou publié

External document link

http://hal.archives-ouvertes.fr/hal-00337661/en/

Date

2010

Journal name

SIAM Journal on Mathematical Analysis

Volume

Number

Publisher

SIAM

Pages

2283-2314

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.

Subjects / Keywords

Smoluchowski's equation; coagulation equation; constant coagulation kernel; self-similar variables; spectral gap; exponential relaxation rate; explicit