On the convergence of the Ohta-Kawasaki Equation to motion by nonlocal Mullins-Sekerka Law

TL;DR

This paper proves that the Ohta-Kawasaki equation converges to a nonlocal Mullins-Sekerka law in smooth domains for dimensions up to three, using Gamma-convergence and interface smoothness assumptions.

Contribution

It introduces a new convergence proof for the Ohta-Kawasaki equation to the nonlocal Mullins-Sekerka law, including a simplified proof for radially symmetric cases.

Findings

Convergence established in dimensions N ≤ 3.

New proof for radially symmetric initial data.

Transport estimates for solutions of the Ohta-Kawasaki equation.

Abstract

In this paper, we establish the convergence of the Ohta-Kawasaki equation to motion by nonlocal Mullins-Sekerka law on any smooth domain in space dimensions $N\leq 3$. These equations arise in modeling microphase separation in diblock copolymers. The only assumptions that guarantee our convergence result are $(i)$ well-preparedness of the initial data and $(ii)$ smoothness of the limiting interface. Our method makes use of the "Gamma-convergence" of gradient flows scheme initiated by Sandier and Serfaty and the constancy of multiplicity of the limiting interface due to its smoothness. For the case of radially symmetric initial data without well-preparedness, we give a new and short proof of the result of M. Henry for all space dimensions. Finally, we establish transport estimates for solutions of the Ohta-Kawasaki equation characterizing their transport mechanism.