The linear Cahn-Hilliard equation with an interface

TL;DR

This paper develops new integral representations for solutions of the linearized Cahn-Hilliard equation with interfaces, using a novel application of Fokas' unified transform method to analyze complex boundary conditions.

Contribution

It introduces a new integral representation approach for the fourth-order Cahn-Hilliard equation with arbitrary initial data and interface conditions, employing a novel implementation of the Fokas method.

Findings

Explicit formulae facilitate analysis of solution properties.

Method handles arbitrary initial data and general interface conditions.

Potential applications in nonlinear well-posedness and free-boundary problems.

Abstract

We obtain new integral representations, expressed as contour integrals in the complex Fourier plane, for the solution of fully nonhomogeneous interface problems for the linearized Cahn-Hilliard equation with arbitrary initial data on the line and general interface conditions prescribed at the origin. Cahn-Hilliard-type models emerge in applied mathematics in connection to a spectacular variety of phenomena of mathematical physics, continuum mechanics, chemistry and biology. A novel implementation of Fokas' unified transform method is in force herein for a fourth-order operator for the first time, with particular challenges arising due to the nature and the generality of the problems under consideration. Our explicit formulae directly lend themselves to exploration of the solution's qualitative properties such as regularity and asymptotic behavior. This work is also useful in the investigation of well-posedness for nonlinear counterparts as well as in the study of free-boundary and diffuse-interface problems.