The Cahn–Hilliard Equation: Recent Advances and Applications

This monograph is based on lectures the author gave at a Conference Board of the Mathematical Sciences - NSF meeting in 2019. Its goal is to describe both classical results and recent developments related to the Cahn-Hilliard equation.  

 

That equation is a fourth-order in space parabolic partial differential equation usually written as 

\( \frac{\partial u}{\partial t}  =  \alpha \kappa \Delta^2 u - \kappa \Delta f(u) \).  It is used to describe the process of phase separation wherein two components of a binary system either spontaneously separate to form separate domains each of which has a pure component, or spontaneously combine to make a binary alloy. The variable \( u \) is an order parameter that represents the normalized relative density of the two components, and \( \kappa \) is a diffusion coefficient. The function \( f \) is the derivative of a potential function with two minima that is used to represent the underlying thermodynamics. While this equation has important uses in material science for understanding mechanical properties like strength, hardness, and ductility, its applications now also include medical and biological ones and - surprisingly - image processing.

 

The book is intended for graduate students and researchers interested in phase separation problems, their generalizations, and applications to other fields. Prior experience with partial differential equations is necessary.

 

The author begins by introducing the basic equation as well as some variations and generalizations. He presents a phenomenological derivation of the equation and then establishes conditions for the existence and uniqueness of weak solutions and their regularity. One chapter focuses on dynamic boundary conditions that are imposed to account for interactions with walls in a confined system. Many of the succeeding sections of the book discuss variations of the basic Cahn-Hilliard equations, but basic results with regular and logarithmic nonlinear terms are treated more extensively. In this case, the associated dynamical system has finite-dimensional global attractors.

 

With the addition of a nonlinear proliferation term, the Cahn-Hilliard equation is used in one of the later chapters to model cells that move, proliferate and interact via diffusion to model biological processes like wound healing or tumor growth.  Another modification of the Cahn-Hilliard equation has been used for image inpainting. This is a kind of interpolation that fills in parts of an image or a video by using information from the surrounding area. Its applications include the restoration of old paintings, removal of scratches or alteration of scenes from photographs, and restoring damaged motion pictures.

 

The author provides an excellent bibliography but unfortunately, there is no index. This is a book for those comfortable with partial differential equations and their analysis, and interested in exploring this equation and its many applications. It also offers useful ideas for potential research projects, especially questions about appropriately modeling and incorporating thermodynamics.

---

 

Bill Satzer (bsatzer@gmail.com), now retired from 3M Company, spent most of his career as a mathematician working in industry on a variety of applications ranging from speech recognition and network modeling to optical films and material science. He did his PhD work in dynamical stems and celestial mechanics.