Regularity of the solution to a nonstandard system of phase field equations

TL;DR

This paper studies the regularity and uniqueness of solutions to a limit system of phase field equations modeling two-species segregation, providing simplified proofs and well-posedness results in a specific framework.

Contribution

It offers a simplified proof of regularity and uniqueness for the limit system of phase field equations, extending prior asymptotic analysis.

Findings

Proved well-posedness of the limit system.

Established regularity of solutions.

Provided a simple proof of uniqueness.

Abstract

A nonstandard system of differential equations describing two-species phase segregation is considered. This system naturally arises in the asymptotic analysis recently done by Colli, Gilardi, Krejci and Sprekels as the diffusion coefficient in the equation governing the evolution of the order parameter tends to zero. In particular, a well-posedness result is proved for the limit system. This paper deals with the above limit problem in a less general but still very significant framework and provides a very simple proof of further regularity for the solution. As a byproduct, a simple uniqueness proof is given as well.