Stationary non-equilibrium solutions for coagulation systems

TL;DR

This paper investigates the existence of stationary solutions in coagulation systems with source terms, revealing that kernel types determine solution existence and providing bounds and asymptotic behaviors.

Contribution

It introduces a comprehensive analysis of stationary solutions for a broad class of coagulation kernels, including new non-existence results for certain kernels.

Findings

Stationary solutions exist for diffusive kernels but not for free molecular kernels.

Derived optimal bounds for large cluster sizes in stationary solutions.

Showed asymptotic equivalence between discrete and continuous coagulation models.

Abstract

We study coagulation equations under non-equilibrium conditions which are induced by the addition of a source term for small cluster sizes. We consider both discrete and continuous coagulation equations, and allow for a large class of coagulation rate kernels, with the main restriction being boundedness from above and below by certain weight functions. The weight functions depend on two power law parameters, and the assumptions cover, in particular, the commonly used free molecular and diffusion limited aggregation coagulation kernels. Our main result shows that the two weight function parameters already determine whether there exists a stationary solution under the presence of a source term. In particular, we find that the diffusive kernel allows for the existence of stationary solutions while there cannot be any such solutions for the free molecular kernel. The argument to prove the non-existence of solutions relies on a novel power law lower bound, valid in the appropriate parameter regime, for the decay of stationary solutions with a constant flux. We obtain optimal lower and upper estimates of the solutions for large cluster sizes, and prove that the solutions of the discrete model behave asymptotically as solutions of the continuous model.