Rate of convergence to self-similarity for Smoluchowski's coagulation equation with constant coefficients
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
TypeArticle accepté pour publication ou publié
External document linkhttp://hal.archives-ouvertes.fr/hal-00337661/en/
Journal nameSIAM Journal on Mathematical Analysis
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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 / KeywordsSmoluchowski's equation; coagulation equation; constant coagulation kernel; self-similar variables; spectral gap; exponential relaxation rate; explicit
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