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

José Cañizo (1) , Stéphane Mischler (2) , Clément Mouhot (2)

1 UAB - Departament de Matemàtiques [Barcelona]
2 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision

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.

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

Domaines: Equations aux dérivées partielles [math.AP]

José Cañizo, Stéphane Mischler, Clément Mouhot. Rate of convergence to self-similarity for Smoluchowski's coagulation equation with constant coefficients. SIAM Journal on Mathematical Analysis, 2010, 41 (6), pp.2283-2314. ⟨10.1137/08074091X⟩. ⟨hal-00337661⟩