Fractional Cahn-Hilliard, Allen-Cahn and porous medium equations

TL;DR

This paper introduces a fractional version of the Cahn-Hilliard equation, proves existence and uniqueness of solutions, explores singular limits leading to fractional Allen-Cahn and porous medium equations, and discusses stationary solutions.

Contribution

It develops a new fractional Cahn-Hilliard model, establishes foundational mathematical properties, and connects it to other fractional PDEs through rigorous limit analysis.

Findings

Existence and uniqueness of weak solutions for the fractional Cahn-Hilliard equation.

Derivation of fractional Allen-Cahn and porous medium equations as singular limits.

Analysis of stationary solutions and their qualitative properties.

Abstract

We introduce a fractional variant of the Cahn-Hilliard equation settled in a bounded domain $\Omega$ of $R^N$ and complemented with homogeneous Dirichlet boundary conditions of solid type (i.e., imposed in the entire complement of $\Omega$). After setting a proper functional framework, we prove existence and uniqueness of weak solutions to the related initial-boundary value problem. Then, we investigate some significant singular limits obtained as the order of either of the fractional Laplacians appearing in the equation is let tend to 0. In particular, we can rigorously prove that the fractional Allen-Cahn, fractional porous medium, and fractional fast-diffusion equations can be obtained in the limit. Finally, in the last part of the paper, we discuss existence and qualitative properties of stationary solutions of our problem and of its singular limits.