Global well-posedness in uniformly local spaces for the Cahn-Hilliard equation in R^3

TL;DR

This paper proves global existence, uniqueness, and regularity of solutions for the Cahn-Hilliard equation in 3D space within uniformly local spaces, including cases with singular potentials and the Cahn-Hilliard-Oono variant.

Contribution

It establishes the first comprehensive results on global well-posedness for infinite-energy solutions in uniformly local spaces for these equations.

Findings

Global existence of solutions for polynomial and singular potentials

Uniqueness and regularity of solutions in the considered spaces

Dissipativity of the solution semigroup for the Cahn-Hilliard-Oono equation

Abstract

We study the infinite-energy solutions of the Cahn-Hilliard equation in the whole 3D space in uniformly local phase spaces. In particular, we establish the global existence of solutions for the case of regular potentials of arbitrary polynomial growth and for the case of sufficiently strong singular potentials. For these cases, the uniqueness and further regularity of the obtained solutions are proved as well. We discuss also the analogous problems for the case of the so-called Cahn-Hilliard-Oono equation where, in addition, the dissipativity of the associated solution semigroup is established.