# Global well-posedness of the 3D Cahn-Hilliard equations

## TL;DR

This paper establishes the global well-posedness of the 3D Cahn-Hilliard equations by constructing approximate solutions, proving their existence and uniqueness, and applying energy estimates, with improved results in a special case.

## Contribution

It provides a rigorous proof of global existence and uniqueness for the 3D Cahn-Hilliard equations, including a refined result for a specific case.

## Key findings

- Existence of solutions via approximate equations and compactness arguments
- Uniqueness of solutions established
- Global well-posedness demonstrated using energy estimates

## Abstract

The Cauchy problem of the Cahn-Hilliard equations is studied in three-dimensional space. Firstly, we construct its approximate fourth-order parabolic equation, obtaining the existence of solutions by the Aubin-Lions's compactness lemma. Furthermore, we prove the uniqueness of the solution. Then, the global well-posedness is demonstrated by using energy estimates. At last, we consider a special case and get a better result about it.