Steady States of Transport-Coagulation-Nucleation Models

TL;DR

This paper investigates the existence and properties of steady states in a complex polymerization model involving transport, nucleation, and coagulation, demonstrating conditions under which steady states exist despite gelation phenomena.

Contribution

It establishes the existence of steady states for a nonlinear integro-differential model with multiplicative coagulation kernels, linking decay rates to steady state properties.

Findings

Steady states exist for certain coagulation kernels despite gelation.

Decay rates of large polymers influence steady state existence.

Numerical experiments illustrate steady state properties.

Abstract

To model the dynamics of polymers formed through nucleation, elongated by polymerisation, shortened by depolymerisation and subject to aggregation reactions, we study a nonlinear integro-differential equation. Growth and shrinkage are described by transport terms, nucleation by a positive boundary condition, and aggregation by a Smoluchowski coagulation kernel. Our main result is the existence of steady states for the multiplicative coagulation kernel despite this kernel producing gelation in finite time for the pure coagulation equation. This is made possible by a sufficiently strong decay rate for large polymers. Beyond the existence result, the qualitative properties of the steady states are illustrated through explicit examples and numerical experiments. The analytical results connect the growth behaviour of the transport velocity and of the coagulation kernel to the decay properties of steady states.