Localization in Stationary Non-equilibrium Solutions for Multicomponent Coagulation Systems

We consider the multicomponent Smoluchowski coagulation equation under non-equilibrium conditions induced either by a source term or via a constant flux constraint. We prove that the corresponding stationary non-equilibrium solutions have a universal localization property. More precisely, we show that these solutions asymptotically localize into a direction determined by the source or by a flux constraint: the ratio between monomers of a given type to the total number of monomers in the cluster becomes ever closer to a predetermined ratio as the cluster size is increased. The assumptions on the coagulation kernel are quite general, with isotropic power law bounds. The proof relies on a particular measure concentration estimate and on the control of asymptotic scaling of the solutions which is allowed by previously derived estimates on the mass current observable of the system.

Automated summary

In this study, the multicomponent Smoluchowski coagulation equation under non-equilibrium conditions is considered, where stationary non-equilibrium solutions are found to have a universal localization property. Specifically, the solutions asymptotically localize into a direction determined by the source or flux constraint as the cluster size increases. The assumptions on the coagulation kernel are quite general with isotropic power law bounds, and the proof relies on measure concentration estimates and control of asymptotic scaling of the solutions.