Macroscopic limit of the Becker-D\"oring equation via gradient flows

TL;DR

This paper extends the gradient flow framework to connect the microscopic Becker-D"oring equation with its macroscopic Lifshitz-Slyozov-Wagner limit, demonstrating convergence of solutions and their gradient structures.

Contribution

It rigorously proves the convergence of both solutions and gradient structures from the Becker-D"oring equation to the coarsening limit, extending previous results.

Findings

Gradient structures for the Becker-D"oring and coarsening equations are established.

Convergence of solutions and gradient structures is rigorously proven.

Small cluster distribution follows a quasistationary distribution influenced by monomer concentration.

Abstract

This work considers gradient structures for the Becker-D\"oring equation and its macroscopic limits. The result of Niethammer [17] is extended to prove the convergence not only for solutions of the Becker-D\"oring equation towards the Lifshitz-Slyozov-Wagner equation of coarsening, but also the convergence of the associated gradient structures. We establish the gradient structure of the nonlocal coarsening equation rigorously and show continuous dependence on the initial data within this framework. Further, on the considered time scale the small cluster distribution of the Becker--D\"oring equation follows a quasistationary distribution dictated by the monomer concentration.