# Kei Fong Lam's research while affiliated with Hong Kong Baptist University and other places

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## Publications (46)

This work concerns a structural topology optimisation problem for 4D printing based on the phase field approach. The concept of 4D printing as a targeted evolution of 3D printed structures can be realised in a two-step process. One first fabricates a 3D object with multi-material active composites and apply external loads in the programming stage....

We investigate the long-time behaviour of strong solutions to a generalised Cahn–Hilliard equation with singular logarithmic potentials and a solution-dependent mass source term. Under appropriate choices of the latter, such models have been used for image inpainting and various biological applications. The logarithmic potential is used to ensure t...

\bfA \bfb \bfs \bft \bfr \bfa \bfc \bft . We study a diffuse interface model describing the evolution of the flow of a binary fluid in a Hele-Shaw cell. The model consists of a Cahn-Hilliard-Darcy type system with transport and mass source. A relevant physical application is related to tumor growth dynamics, which in particular justifies the occurr...

A phase field approach for structural topology optimization with application to additive manufacturing is analyzed. The main novelty is the penalization of overhangs (regions of the design that require underlying support structures during construction) with anisotropic energy functionals. Convex and non-convex examples are provided, with the latter...

In this paper, we study an optimal control problem for a macroscopic mechanical tumor model based on the phase field approach. The model couples a Cahn-Hilliard-type equation to a system of linear elasticity and a reaction-diffusion equation for a nutrient concentration. By taking advantage of previous analytical well-posedness results established...

We extend previous weak well-posedness results obtained in Frigeri et al. (2017, Solvability, Regularity, and Optimal Control of Boundary Value Problems for PDEs , Vol. 22, Springer, Cham, pp. 217–254) concerning a non-local variant of a diffuse interface tumour model proposed by Hawkins-Daarud et al. (2012, Int. J. Numer. Method Biomed. Engng. 28...

Mechanical effects have mostly been neglected so far in phase field tumour models that are based on a Cahn–Hilliard approach. In this paper we study a macroscopic mechanical model for tumour growth in which cell–cell adhesion effects are taken into account with the help of a Ginzburg–Landau type energy. In the overall model an equation of Cahn–Hill...

The Cahn–Hilliard equation is one of the most common models to describe phase separation processes of a mixture of two materials. For a better description of short-range interactions between the material and the boundary, various dynamic boundary conditions for the Cahn–Hilliard equation have been proposed and investigated in recent times. Of parti...

In this paper, we study an optimal control problem for a macroscopic mechanical tumour model based on the phase field approach. The model couples a Cahn--Hilliard type equation to a system of linear elasticity and a reaction-diffusion equation for a nutrient concentration. By taking advantage of previous analytical well-posedness results establishe...

We study a diffuse interface model describing the evolution of the flow of a binary fluid in a Hele-Shaw cell. The model consists of a Cahn-Hilliard-Darcy (CHD) type system with transport and mass source. A relevant physical application is related to tumor growth dynamics, which in particular justifies the occurrence of a mass inflow. We study the...

We prove the existence of unique weak solutions to an extension of a Cahn–Hilliard model proposed recently by C Liu and H Wu (2019 Arch. Ration. Mech. Anal. 233 167–247), in which the new dynamic boundary condition is further generalised with an affine linear relation between the surface and bulk phase field variables. As a first approach to tackle...

We consider the inverse problem of identifying parameters in a variant of the diffuse interface model for tumour growth model proposed by Garcke, Lam, Sitka and Styles (Math. Models Methods Appl. Sci. 2016). The model contains three constant parameters; namely the tumour growth rate, the chemotaxis parameter and the nutrient consumption rate. We st...

We study a phase field model proposed recently in the context of tumour growth. The model couples a Cahn–Hilliard–Brinkman (CHB) system with an elliptic reaction-diffusion equation for a nutrient. The fluid velocity, governed by the Brinkman law, is not solenoidal, as its divergence is a function of the nutrient and the phase field variable, i.e.,...

An inverse problem of reconstructing the magnetic reluctivity in a quasilinear magnetostatic Maxwell system is studied. To overcome the ill-posedness of the inverse problem, we propose and investigate two regularisations posed as constrained minimisation problems. The first uses the total variation (perimeter) regularisation, and the second makes u...

The Cahn--Hilliard equation is one of the most common models to describe phase separation processes of a mixture of two materials. For a better description of short-range interactions between the material and the boundary, various dynamic boundary conditions for the Cahn--Hilliard equation have been proposed and investigated in recent times. Of par...

A diffuse interface model for surfactants in multi-phase flow with three or more fluids is derived. A system of Cahn–Hilliard equations is coupled with a Navier-Stokes system and an advection-diffusion equation for the surfactant ensuring thermodynamic consistency. By an asymptotic analysis the model can be related to a moving boundary problem in t...

Mechanical effects have mostly been neglected so far in phase field tumour models that are based on a Cahn-Hilliard approach. In this paper we study a macroscopic mechanical model for tumour growth in which cell-cell adhesion effects are taken into account with the help of a Ginzburg--Landau type energy. In the overall model an equation of Cahn-Hil...

We study a phase field model proposed recently in the context of tumour growth. The model couples a Cahn-Hilliard-Brinkman (CHB) system with a elliptic reaction-diffusion equation for a nutrient. The fluid velocity, governed by the Brinkman law, is not solenodial, as its divergence is a function of the nutrient and the phase field variable, i.e., s...

We prove the existence of unique weak solutions to an extension of a Cahn--Hilliard model proposed recently by C.~Liu and H.~Wu (2019), in which the new dynamic boundary condition is further generalised with an affine linear relation between the surface and bulk order parameters. As a first approach to tackle more general and nonlinear relations, w...

We study the sharp interface limit of a non-mass-conserving Cahn--Hilliard--Darcy system with the weak compactness method developed in Chen (J. Differential Geometry, 1996). The source term present in the Cahn--Hilliard component is a product of the order parameter and a prescribed function with zero spatial mean, leading to non-conservation of mas...

We consider a coupled bulk--surface Allen--Cahn system affixed with a Robin-type boundary condition between the bulk and surface variables. This system can also be viewed as a relaxation to a bulk--surface Allen--Cahn system with non-trivial transmission conditions. Assuming that the nonlinearities are real analytic, we prove the convergence of eve...

A diffuse interface model for surfactants in multi-phase flow with three or more fluids is derived. A system of Cahn-Hilliard equations is coupled with a Navier-Stokes system and an advection-diffusion equation for the surfactant ensuring thermodynamic consistency. By an asymptotic analysis the model can be related to a moving boundary problem in t...

We consider the shape optimization of an object in Navier--Stokes flow by employing a combined phase field and porous medium approach, along with additional perimeter regularization. By considering integral control and state constraints, we extend the results of earlier works concerning the existence of optimal shapes and the derivation of first or...

Topology changes in multi-phase fluid flows are difficult to model within a traditional sharp interface theory. Diffuse interface models turn out to be an attractive alternative to model two-phase flows. Based on a Cahn-Hilliard-Navier-Stokes model introduced by Abels, Garcke and Gr\"{u}n (Math. Models Methods Appl. Sci. 2012), which uses a volume...

In this paper we consider the identification of internal parameters in a diffuse interface model for tumour growth. The tumour is modelled by a phase field that acts as a smooth indicator function to distinguish the healthy cells and the tumour cells inside the tissue. Our model is a variant of the model proposed in [Garcke et al., Math. Models Met...

In this paper we consider the identification of internal parameters in a diffuse interface model for tumour growth. The tumour is modelled by a phase field that acts as a smooth indicator function to distinguish the healthy cells and the tumour cells inside the tissue. Our model is a variant of the model proposed in [Garcke et al., Math. Models Met...

We study the existence of weak solutions to a mixture model for tumour growth that consists of a Cahn–Hilliard–Darcy system coupled with an elliptic reaction-diffusion equation. The Darcy law gives rise to an elliptic equation for the pressure that is coupled to the convective Cahn–Hilliard equation through convective and source terms. Both Dirichl...

We study a coupled bulk-surface Allen-Cahn system with an affine linear transmission condition, that is, the trace values of the bulk variable and the values of the surface variable are connected via an affine relation, and this serves to generalize the usual dynamic boundary conditions. We tackle the problem of well-posedness via a penalization me...

The inpainting of damaged images has a wide range of applications and many different mathematical methods have been proposed to solve this problem. Inpainting witht the help of Cahn--Hilliard models has been particularly successful, and it turns out that Cahn--Hilliard inpainting with the double obstacle potential can lead to better results compare...

We consider a model describing the evolution of a tumor inside a host tissue in terms of the parameters $\varphi_p$, $\varphi_d$ (proliferating and dead cells, respectively), $u$ (cell velocity) and $n$ (nutrient concentration). The variables $\varphi_p$, $\varphi_d$ satisfy a Cahn-Hilliard type system with nonzero forcing term (implying that their...

We study a non-local variant of a diffuse interface model proposed by Hawkins–Daarud et al. (Int. J. Numer. Methods Biomed. Eng. 28:3–24, 2012) for tumour growth in the presence of a chemical species acting as nutrient. The system consists of a Cahn–Hilliard equation coupled to a reaction-diffusion equation. For non-degenerate mobilities and smooth...

New diffuse interface and sharp interface models for soluble and insoluble surfactants fulfilling energy inequalities are introduced. We discuss their relation with the help of asymptotic analysis and present an existence result for a particular diffuse interface model.

We study a non-local variant of a diffuse interface model proposed by Hawkins--Darrud et al. (2012) for tumour growth in the presence of a chemical species acting as nutrient. The system consists of a Cahn--Hilliard equation coupled to a reaction-diffusion equation. For non-degenerate mobilities and smooth potentials, we derive well-posedness resul...

We derive a class of Navier--Stokes--Cahn--Hilliard systems that models two-phase flows with mass transfer coupled to the process of chemotaxis. These thermodynamically consistent models can be seen as the natural Navier--Stokes analogues of earlier Cahn--Hilliard--Darcy models proposed for modeling tumor growth, and are derived based on a volume-a...

We derive a Cahn-Hilliard-Darcy model to describe multiphase tumour growth taking interactions with multiple chemical species into account as well as the simultaneous occurrence of proliferating, quiescent and necrotic regions. Via a coupling of the Cahn-Hilliard-Darcy equations to a system of reaction-diffusion equations a multitude of phenomena s...

We study the existence of weak solutions to a mixture model for tumour growth that consists of a Cahn--Hilliard--Darcy system coupled with an elliptic reaction-diffusion equation. The Darcy law gives rise to an elliptic equation for the pressure that is coupled to the convective Cahn--Hilliard equation through convective and source terms. Both Diri...

New diffuse interface and sharp interface models for soluble and insoluble surfactants fulfilling energy inequalities are introduced. We discuss their relation with the help of asymptotic analysis and present an existence result for a particular diffuse interface model.

We study the existence of weak solutions to a Cahn--Hilliard--Darcy system coupled with a convection-reaction-diffusion equation through the fluxes, through the source terms and in Darcy's law. The system of equations arises from a mixture model for tumour growth accounting for transport mechanisms such as chemotaxis and active transport. We prove,...

We consider a diffuse interface model for tumor growth consisting of a Cahn--Hilliard equation with source terms coupled to a reaction-diffusion equation, which models a tumor growing in the presence of a nutrient species and surrounded by healthy tissue. The well-posedness of the system equipped with Neumann boundary conditions was found to requir...

We consider a diffuse interface model for tumour growth consisting of a
Cahn--Hilliard equation with source terms coupled to a reaction-diffusion
equation. The coupled system of partial differential equations models a tumour
growing in the presence of a nutrient species and surrounded by healthy tissue.
The model also takes into account transport m...

Using basic thermodynamic principles we derive a Cahn--Hilliard--Darcy model
for tumour growth including nutrient diffusion, chemotaxis, active transport,
adhesion, apoptosis and proliferation. The model generalises earlier models and
in particular includes active transport mechanisms which ensure thermodynamic
consistency. We perform a formally ma...

We consider shape and topology optimization for fluids which are governed by
the Navier--Stokes equations. Shapes are modelled with the help of a phase
field approach and the solid body is relaxed to be a porous medium. The phase
field method uses a Ginzburg--Landau functional in order to approximate a
perimeter penalization. We focus on surface fu...

We analyse a diffuse interface type approximation, known as the diffuse
domain approach, of a linear coupled bulk-surface elliptic partial differential
system. The well-posedness of the diffuse domain approximation is shown using
weighted Sobolev spaces and we prove that the solution to the diffuse domain
approximation converges weakly to the solut...

Phase field models for two-phase flow with a surfactant soluble in possibly
both fluids are derived from balance equations and an energy inequality so that
thermodynamic consistency is guaranteed. Via a formal asymptotic analysis, they
are related to sharp interface models. Both cases of dynamic as well as
instantaneous adsorption are covered. Flex...

## Citations

... Let us mention that the system analyzed in [12] (see also [22,24]) readily frames into (6.2)-(6.5) (neglecting the velocity field and using a simplified version of (6.16)) where g = −χn, S(ϕ) = S(ϕ, n) = −mϕ + h(ϕ, n). ...

... The techniques developed in this paper allow us to extend some previous well-posedness results known for Cahn-Hilliard equations with source terms (see, for instance, [9,26,28,29,34]). In particular, our approach is more general compared to the current state of arts as it allows the initial datum ϕ 0 to be also the pure phase , say, 0, and this does not comply with the standard requirement (ϕ 0 ) Ω ∈ (0, 1). ...

... The investigation of associated optimal control problems also presents a wide number of results of which we mention [9,12,15,20,21,26,32,35,40,[43][44][45]47,48]. The optimal control problem (CP) has recently been investigated by the present authors in [16] for the case of regular or logarithmic nonlinearities F 1 . ...

... For some nonlocal variations of the above model we refer to [25,26,42]. Moreover, in order to better emulate in-vivo tumor growth, it is possible to include in similar models the effects generated by the fluid flow development by postulating a Darcy's law or a Stokes-Brinkman's law. ...

... In particular, when one considers surface dynamics, an issue emerges right away that is what is the relation between the phase field variable in the bulk and the one on the boundary? By assuming these two are the same at the boundary, a series of studies have examined the issue of thermodynamically consistency [23][24][25][26][27]. By not assuming that they coincide on the boundary, Knopf et al. derived a set of boundary conditions for the Cahn-Hilliard model (known as the Knopf-Lam model) in 2020 [28] and for non-local models (known as the Knopf-Signori model) in 2021 [29]. ...

... In recent years, this kind of systematic research has attracted many mathematicians and has made remarkable progress [4][5][6][7][8]. In order to overcome the special difficulties brought by nonlinear terms, many new ideas and tools have been developed, which greatly enrich the theory of partial differential equations [9][10][11][12][13][14]. ...

... Rigorous analysis of the coupling of the twophase Cahn-Hilliard theory with incompressible fluid flows on the basis of Korteweg stress tensor dynamics have been investigated by many authors (see, e.g., [2, 3, 18-21, 24, 25, 27-30] and the references therein, just to give some examples of the most important developments), resulting in a fairly reasonable theoretical picture about the corresponding binary fluid behavior in most cases. Coupling of hydrodynamic models for fluid behavior with a N -component family of Cahn-Hilliard models has also been considered recently 1 (see, e.g., [5,14,31]). As such, existence of weak solutions to a class of N -component Cahn-Hilliard systems, subject to a mechanism of cross-diffusion between different chemical species and singular bulk potentials, has recently been studied in [16]. ...

... Moreover, in order to better emulate in-vivo tumor growth, it is possible to include in similar models the effects generated by the fluid flow development by postulating a Darcy's law or a Stokes-Brinkman's law. In this direction, we refer to [19,22,25,[27][28][29][30][31]33,51], and we also mention [34], where elastic effects are included. For further models, discussing the case of multispecies, we refer the reader to [19,25]. ...

... By assuming these two are the same at the boundary, a series of studies have examined the issue of thermodynamically consistency [23][24][25][26][27]. By not assuming that they coincide on the boundary, Knopf et al. derived a set of boundary conditions for the Cahn-Hilliard model (known as the Knopf-Lam model) in 2020 [28] and for non-local models (known as the Knopf-Signori model) in 2021 [29]. The latter studies opened up a new venue for one to examine the thermodynamically consistency in the bulk and surface dynamics holistically. ...

... In order to derive a more manageable problem, the perimeter functional was relaxed using a phase-field approach justified by showing the Γ-convergence of the relaxed functional to the functional with perimeter penalization. In recent years this kind of approach has been successfully implemented in inverse boundary value problems for partial differential equations and systems, see for example [12], [41], [53], [44], [30]. ...