Localization in stationary non-equilibrium solutions for multicomponent coagulation systems

TL;DR

This paper proves that stationary solutions of multicomponent coagulation systems under non-equilibrium conditions tend to localize in a specific direction, with ratios of monomer types stabilizing as cluster size grows.

Contribution

It establishes a universal localization property for non-equilibrium solutions of multicomponent Smoluchowski coagulation equations with general isotropic kernels.

Findings

Solutions asymptotically localize in a direction determined by source or flux.

Ratios of monomer types approach a predetermined ratio for large clusters.

The proof uses measure concentration and asymptotic scaling control.

Abstract

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.