Infinite energy solutions for the Cahn-Hilliard equation in cylindrical domains

TL;DR

This paper investigates infinite-energy solutions of the 3D Cahn-Hilliard equation in cylindrical domains, establishing well-posedness, dissipativity, regularity, and attractor existence for various potentials, including singular ones.

Contribution

It provides new results on the existence, regularity, and long-term behavior of solutions for the Cahn-Hilliard equation in cylindrical domains, covering both regular and singular potentials.

Findings

Well-posedness and dissipativity established for regular potentials.

Existence of global attractors for regular potentials.

Trajectory attractors identified for non-unique cases like logarithmic potentials.

Abstract

We give a detailed study of the infinite-energy solutions of the Cahn-Hilliard equation in the 3D cylindrical domains in uniformly local phase space. In particular, we establish the well-posedness and dissipativity for the case of regular potentials of arbitrary polynomial growth as well as for the case of sufficiently strong singular potentials. For these cases, we prove the further regularity of solutions and the existence of a global attractor. For the cases where we have failed to prove the uniqueness (e.g., for the logarithmic potentials), we establish the existence of the trajectory attractor and study its properties.