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Project Details
Funding Scheme : General Research Fund
Project Number : 14302319
Project Title(English) : On Cahn-Hilliard models with singular potentials and source terms 
Project Title(Chinese) : 關於具有奇異勢函數和源項的Cahn-Hilliard模型 
Principal Investigator(English) : Prof Lam, Kei Fong 
Principal Investigator(Chinese) :  
Department : Department of Mathematics
Institution : Hong Kong Baptist University
E-mail Address : akflam@hkbu.edu.hk 
Tel :  
Co - Investigator(s) :
Panel : Physical Sciences
Subject Area : Mathematics
Exercise Year : 2019 / 20
Fund Approved : 502,444
Project Status : Completed
Completion Date : 28-2-2023
Project Objectives :
We shall investigate the existence of stationary solutions to the Cahn-Hilliard model with singular potentials and inpainting source terms. Then, we shall extend the result to account for the stationary problem and the time-dependent problem with more general source terms.
We shall develop iterative numerical schemes based on the existence proof in Objective 1 to solve the stationary Cahn-Hilliard inpainting model with singular potentials. Then, we shall investigate how to implement the numerical schemes and conduct numerical experiments to compare with previous methods for inpainting.
We shall investigate a Cahn-Hilliard model with singular potentials and tumour growth source terms. We propose to derive regularity estimates for an auxiliary variable, from which the regularity estimates for the solution variables can be obtained. Then, we shall use these new regularity estimates to establish the uniqueness of solutions to the tumour model.
We shall extend the results of Objective 3 to a multiphase Cahn-Hilliard model with singular potentials and tumour growth source terms. We propose to identify suitable auxiliary variables whose regularity properties can be derived more easily. Then, following a similar strategy, we shall develop new regularity arguments for the solution to the multiphase tumour model and establish the uniqueness of solutions.
Abstract as per original application
(English/Chinese):		 The Cahn-Hilliard (CH) equation is a fourth order nonlinear partial differential equation (PDE) originally used to model the separation of a mixture containing two phases of matter, such as water and oil, and is now an ubiquitous component of many mathematical models in applied sciences. It is formulated with a nonconvex potential function, usually taken as a smooth polynomial, that possesses two equal minima, say -1 and 1. Through the addition of appropriate source terms, the CH equation has seen recent applications in modelling the growth of tumours and in repairing damaged black-and-white images (also known as inpainting). However, an unfortunate feature with a polynomial potential is that the solution may not stay bounded in between -1 and 1, and undesirable effects, such as negative mass densities in tumour models or new shades of colour in black-and-white images, can occur. A remedy is to employ singular potentials, such as a logarithmic-type function, to ensure that the solution to the model stays in between -1 and 1. In exchange, the mathematical analysis of these models with singular potentials become more involved compared to the polynomial potentials. For recent applications in tumour growth and inpainting, the new combination of singular potentials and source terms has not received much attention in the literature, and current analytical studies are mostly confined to establishing the existence of weak solutions. In this project, we plan to expand the scope of the analysis for these new Cahn-Hilliard models with singular potentials and source terms by investigating the issues of uniqueness and regularity of solutions, as well as addressing the existence of stationary solutions analytically and also numerically. We believe that the proposed methodologies can yield both theoretical and practical contributions towards the active research areas of tumour growth and inpainting, in the form of a novel and alternative strategy to perform inpainting with the CH equation that is built upon a rigorous analytical basis, and a mathematical foundation for practitioners to make meaningful comparisons between tumour model simulations and experimental data.
Cahn-Hilliard（CH）方程是一個四階非線性偏微分方程，最初用於模擬分離含有兩相物質（例如水和油）的混合物，現在是在應用科學許多數學模型中普遍存在的組成部分。它通常包含一個非凸勢函數，具有兩個相等的最小值，比如 -1 和1。通過添加適當的源項，最近CH方程已經應用於模擬腫瘤的生長和修復受損的黑白圖像。然而，具有源項的不幸特徵是該解決方案可能不會保持在-1和1之間，並且具有不期望的效果，例如在腫瘤模型中出現負質量密度或在修復受損的黑白圖像出現新的灰色地帶。 一種補救措施是採用奇異勢函數，以確保模型的解決方案保持在-1和1之間。作為交換，這些具有奇異勢函數的模型的數學分析變得更加複雜。對於腫瘤生長和修復受損圖像這兩種最新應用，這個奇異勢函數和源項的新組合在文獻中並未引起太多關注，目前的分析研究主要局限於建立弱解方案的存在性。 在這個項目中，我們計劃通過研究解決方案獨特性和規律性的問題，以及通過分析和數字方式解決固定解決方案的存在，擴展這些具有奇異勢函數和源項的新型CH模型的分析範圍 。我們相信，所提出的方法可以為腫瘤生長和修復受損圖像的活躍研究領域提供理論和實踐的貢獻，例如新的圖像修復策略建立在嚴格的分析基礎上，以及為從業者提供數學基礎，以便在腫瘤模型模擬和實驗數據之間進行有意義的比較。
Realisation of objectives: All project objectives have been achieved.
Summary of objectives addressed:
Objectives Addressed Percentage achieved
1.We shall investigate the existence of stationary solutions to the Cahn-Hilliard model with singular potentials and inpainting source terms. Then, we shall extend the result to account for the stationary problem and the time-dependent problem with more general source terms.Yes100%
2.We shall develop iterative numerical schemes based on the existence proof in Objective 1 to solve the stationary Cahn-Hilliard inpainting model with singular potentials. Then, we shall investigate how to implement the numerical schemes and conduct numerical experiments to compare with previous methods for inpainting.Yes100%
3.We shall investigate a Cahn-Hilliard model with singular potentials and tumour growth source terms. We propose to derive regularity estimates for an auxiliary variable, from which the regularity estimates for the solution variables can be obtained. Then, we shall use these new regularity estimates to establish the uniqueness of solutions to the tumour model.Yes100%
4.We shall extend the results of Objective 3 to a multiphase Cahn-Hilliard model with singular potentials and tumour growth source terms. We propose to identify suitable auxiliary variables whose regularity properties can be derived more easily. Then, following a similar strategy, we shall develop new regularity arguments for the solution to the multiphase tumour model and establish the uniqueness of solutions.Yes100%
Research Outcome
Major findings and research outcome: We studied several Cahn-Hilliard models with singular potentials and source terms that have seen applications in image processing and tumour growth. Through a series of auxiliary results developed to handle the interplay between the potentials and relevant source terms, our chief contributions are the existence of weak solutions, well-posedness of strong solutions, long-time behaviour in the context of global and exponential attractors, optimal control and parameter identification for the models, as well as stable and convergent numerical schemes. In particular, our research result is the first in obtaining a direct proof of stationary solutions of these kind of systems that do not admit an obvious Lyapunov structure. These works are published in some of the best applied mathematics journals, such as SIAM Journal on Mathematical Analysis, Journal of Differential Equations, SIAM Journal on Control and Optimization, Advances in Nonlinear Analysis and Journal of Numerical Mathematics.
Potential for further development of the research
and the proposed course of action:		 While the main focus of the project is around the existence and uniqueness of solutions to the associated Cahn-Hilliard models, practitioners are more interested in optimisation applications. In the context of tumour modelling these involve parameter estimation, computing optimal dosage of chemotherauptic drugs and optimal treatment times. In this project we have several analytical results but did not explore the numerical aspects in part due to the lack of data. Hence, it would be very interesting to combine our analytical investigations with the clinical data available to the practitioners in developing some form of data-driven cancer treatment based on these Cahn-Hilliard models. These topics would make very good PhD research projects.
Layman's Summary of
Completion Report:		 The Cahn-Hilliard equation appears in many areas of scientific applications, ranging from fluid dynamics, material sciences, to image processing and tumour modelling, and so it is essential to develop mathematical models that can accurately capture the relevant physical phenomena. This will involve singular terms that lead to a challenging mathematical study. In this project we explored the mathematical analysis of realistic models based on the Cahn-Hilliard equation that are currently in use by practitioners for tumour growth. Our theoretical investigations demonstrate that these models are mathematically sound, leading to further studies in related optimal control problems and the development of convergent numerical schemes. Our ideas developed to tackle the challenging singular terms can also be used on similar types of equations prevalent in other areas of scientific applications, such as pattern formation and active emulsions.
Research Output
Peer-reviewed journal publication(s)
arising directly from this research project :
(* denotes the corresponding author)
Year of
Publication		 Author(s) Title and Journal/Book Accessible from Institution Repository
2020 Matthias Ebenbeck and Kei Fong Lam*  Weak and stationary solutions to a Cahn-Hilliard-Brinkman model with singular potentials and source terms.  No 
2022 Sergio Frigeri, Kei Fong Lam, Andrea Signori*  Strong well-posedness and inverse identification problem of a non-local phase field tumor model with degenerate mobilities  No 
2022 Andrea Giorgini, Kei Fong Lam, Elisabetta Rocca, Giulio Schimperna*  On the Existence of Strong Solutions to the Cahn--Hilliard--Darcy System with Mass Source  No 
2021 Harald Garcke, Kei Fong Lam*, Andrea Signori  Sparse Optimal Control of a Phase Field Tumor Model with Mechanical Effects  No 
2022 Kei Fong Lam*  Global and exponential attractors for a Cahn–Hilliard equation with logarithmic potentials and mass source  No 
Kei Fong Lam* and Ru Wang  Stability and convergence of relaxed scalar auxiliary variable schemes for Cahn–Hilliard systems with bounded mass source  No 
Recognized international conference(s)
in which paper(s) related to this research
project was/were delivered :
Month/Year/City Title Conference Name
Online Global and Exponential Attractors for a Cahn-Hilliard Equation with Logarithmic Potentials and Mass Source  SIAM Conference on Applications of Dynamical Systems 
Wilmington, NC Scalar auxiliary variable schemes for Cahn-Hilliard systems with mass source  The 13th AIMS International Conference on Dynamical Systems, Differential Equations and Applications 
Tokyo Scalar auxiliary variable schemes for Cahn-Hilliard systems with mass source  ICIAM Tokyo 2023 
Other impact
(e.g. award of patents or prizes,
collaboration with other research institutions,
technology transfer, etc.):
  SCREEN ID: SCRRM00542