Global existence for the continuous nonhomogeneous Smoluchowski equation and asymptotic behaviour of the solutions
type: Item
instance of: scholarly article
title: Global existence for the continuous nonhomogeneous Smoluchowski equation and asymptotic behaviour of the solutions
zbMATH Open document ID: 1036.35072
DOI: 10.1016/S1631-073X(03)00070-0
author: Stéphane Mischler
author: Mariano Rodríguez Ricard
published in: Comptes Rendus. Mathématique. Académie des Sciences, Paris
publication date: 2003-09-15T00:00:00Z
review text: Global existence of weak solutions to the continuous Smoluchowski coagulation equation with diffusion is established for coagulation kernels \(a\) satisfying the structure condition \(a(y,y')\leq a(y,y+y') + a(y',y+y')\), \((y,y')\in (0,\infty)^2\). This assumption is a generalization of the monotonicity condition \(a(y,y')\leq a(y,y+y')\) used previously, and includes several coagulation kernels which could not be considered with the latter, particularly Smoluchowski's coagulation kernel \((y^{1/3} + {y'}^{1/3}) (y^{-1/3} +{ y'}^{-1/3})\). The existence proof relies on the fact that, under this structure condition, the \(L^p\)-norms of the solutions are non-increasing functions of time, while the dissipation rate of the \(L^p\)-norm provides a weak compactness estimate in \(L^1\) for the coagulation reaction term.
reviewed by: Philippe Laurençot
Mathematics Subject Classification ID: 35K45
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35B45
zbMATH DE Number: 1981263
zbMATH Keywords: Smoluchowski's coagulation equation
zbMATH Keywords: Lyapunov functional
zbMATH Keywords: weak compactness
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1981263