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 particular interests are the model by Goldstein, Miranville and Schimperna (Physica D, 2011) and the model by Liu and Wu (Arch.~Ration.~Mech.~Anal., 2019). Both of these models satisfy similar physical properties but differ greatly in their mass conservation behaviour. In this paper we introduce a new model which interpolates between these previous models, and investigate analytical properties such as the existence of unique solutions and convergence to the previous models mentioned above in both the weak and the strong sense. For the strong convergences we also establish rates in terms of the interpolation parameter, which are supported by numerical simulations obtained from a fully discrete, unconditionally stable and convergent finite element scheme for the new interpolation model.
We prove energy stability of a standard operator-splitting method for the Cahn-Hilliard equation. We establish uniform bound of Sobolev norms of the numerical solution and convergence of the splitting approximation. This is the first unconditional energy stability result for the operator-splitting method for the Cahn-Hilliard equation. Our analysis can be extended to many other models.
Cahn-Hilliard models are central for describing the evolution of interfaces in phase separation processes and free boundary problems. In general, they have non-constant and often degenerate mobilities. However, in the latter case, the spontaneous appearance of points of vanishing mobility and their impact on the solution are not well understood. In this paper we develop a singular perturbation theory to identify a range of degeneracies for which the solution of the Cahn-Hilliard equation forms a singularity in infinite time. This analysis forms the basis for a rigorous sharp interface theory a…
We consider Cahn-Hilliard equations with external forcing terms. Energy decreasing and mass conservation might not hold. We show that level surfaces of the solutions of such generalized Cahn-Hilliard equations tend to the solutions of a moving boundary problem under the assumption that classical solutions of the latter exist. Our strategy is to construct approximate solutions of the generalized Cahn-Hilliard equation by the Hilbert expansion method used in kinetic theory and proposed for the standard Cahn-Hilliard equation, by Carlen, Carvalho and Orlandi, \cite {CCO}. The constructed approxim…
This paper tackles the approximation of surface diffusion flow using a Cahn--Hilliard-type model. We introduce and analyze a new second order variational phase field model which associates the classical Cahn--Hilliard energy with two degenerate mobilities. This association allows to gain an order of approximation of the sharp limit. In a second part, we propose some simple and efficient numerical schemes to approximate the solutions, and we provide numerical 2D and 3D experiments that illustrate the interest of our model in comparison with other Cahn--Hilliard models.
We consider a Strang-type second order operator-splitting discretization for the Cahn-Hilliard equation. We introduce a new theoretical framework and prove uniform energy stability of the numerical solution and persistence of all higher Sobolev norms. This is the first strong stability result for second order operator-splitting methods for the Cahn-Hilliard equation. In particular we settle several long-standing open issues in the work of Cheng, Kurganov, Qu and Tang \cite{Tang15}.
In this note, we want to highlight and correct an error in the paper "On the nonlocal Cahn-Hilliard-Brinkman and Cahn-Hilliard-Hele-Shaw systems" [Comm. Pure Appl. Anal. 15 (2016), 299-317] written by the authors.
We consider the two-dimensional Cahn-Hilliard equation with logarithmic potentials and periodic boundary conditions. We employ the standard semi-implicit numerical scheme which treats the linear fourth-order dissipation term implicitly and the nonlinear term explicitly. Under natural constraints on the time step we prove strict phase separation and energy stability of the semi-implicit scheme. This appears to be the first rigorous result for the semi-implicit discretization of the Cahn-Hilliard equation with singular potentials.
In the current work, we are performing the asymptotic analysis, beyond the periodic setting, of the Cahn-Hilliard-Navier-Stokes system. Under the general deterministic distribution assumption on the microstructures in the domain, we find the limit model equivalent to the heterogeneous one. To this end, we use the sigma-convergence concept which is suitable for the passage to the limit.
We study the well-posedness of a modified degenerate Cahn-Hilliard type model for surface diffusion. With degenerate phase-dependent diffusion mobility and additional stabilizing function, this model is able to give the correct sharp interface limit. We introduce a notion of weak solutions for the nonlinear model. The existence result is obtained by approximations of the proposed model with nondegenerate mobilities. We also employ this method to prove existence of weak solutions to a related model where the chemical potential contains a nonlocal term originated from self-climb of dislocations …
Over the last decades, many diffuse-interface Navier-Stokes Cahn-Hilliard models with non-matching densities have appeared in the literature. These models claim to describe the same physical phenomena, yet they are distinct from one another. The overarching objective of this work is to bring all of these models together by laying down a unified framework of Navier-Stokes Cahn-Hilliard models with non-zero mass fluxes. Our development is based on three unifying principles: (1) there is only one system of balance laws based on continuum mixture theory that describes the physical model, (2) there…
We investigate a multiphase Cahn-Hilliard model for tumor growth with general source terms. The multiphase approach allows us to consider multiple cell types and multiple chemical species (oxygen and/or nutrients) that are consumed by the tumor. Compared to classical two-phase tumor growth models, the multiphase model can be used to describe a stratified tumor exhibiting several layers of tissue (e.g., proliferating, quiescent and necrotic tissue) more precisely. Our model consists of a convective Cahn-Hilliard type equation to describe the tumor evolution, a velocity equation for the associat…
We give multiplicity results for the solutions of a nonlinear elliptic equation, with an asymmetric double well potential of Van der Waals-Allen--Cahn--Hilliard type, satisfying a linear volume constraint, on a bounded Lipschitz domain $Ω\subset\mathds R^N$. The number of solutions is estimated in terms of topological and homological invariants of the underlying domain $Ω$.
The aim of this paper is to study the well-posedness and the existence of global attractors for a family of Cahn-Hilliard equations with a mobility depending on the chemical potential. Such models arise from generalizations of the (classical) Cahn-Hilliard equation due to M. E. Gurtin.
This note completely resolves the asymptotic development of order $2$ by $Γ$-convergence of the mass-constrained Cahn--Hilliard functional, by showing that one of the critical assumptions of the authors' previous work (Leoni, Murray, Second-order $Γ$-limit for the Cahn--Hilliard functional, Arch. Ration. Mech. Anal. 219, 3, 2016) is unnecessary.
In this paper, on the basis of the Schauder type estimates and Campanato spaces, we prove the global existence of classical solutions for a generalized Cahn-Hilliard equation with biological applications.
A boundary control problem for the viscous Cahn-Hilliard equations with possibly singular potentials and dynamic boundary conditions is studied and first order necessary conditions for optimality are proved. Key words: Cahn-Hilliard equation, dynamic boundary conditions, phase separation, singular potentials, optimal control, optimality conditions, adjoint state system
In this paper we study a nonlocal Cahn-Hilliard model (CHE) in the framework of random walk spaces, which includes as particular cases, the CHE on locally finite weighted connected graphs, the CHE determined by finite Markov chains or the Cahn-Hilliard Equations driven by convolution integrable kernels. We consider different transitions for the phase and the chemical potential, and a large class of potentials including obstacle ones. We prove existence and uniqueness of solutions in $L^1$ of the Cahn-Hilliard Equation. We also show that the Cahn-Hilliard equation is the gradient flow of the Gi…
An asymptotic limit of a class of Cahn-Hilliard systems is investigated to obtain a general nonlinear diffusion equation. The target diffusion equation may reproduce a number of well-known model equations: Stefan problem, porous media equation, Hele-Shaw profile, nonlinear diffusion of singular logarithmic type, nonlinear diffusion of Penrose-Fife type, fast diffusion equation and so on. Namely, by setting the suitable potential of the Cahn-Hilliard systems, all of these problems can be obtained as limits of the Cahn-Hilliard related problems. Convergence results and error estimates are proved…
A boundary control problem for the pure Cahn-Hilliard equations with possibly singular potentials and dynamic boundary conditions is studied and first-order necessary conditions for optimality are proved. Key words: Cahn-Hilliard equation, dynamic boundary conditions, phase separation, singular potentials, optimal control, optimality conditions
We prove existence, uniqueness, regularity and separation properties for a nonlocal Cahn-Hilliard equation with a reaction term. We deal here with the case of logarithmic potential and degenerate mobility as well an uniformly lipschitz in $u$ reaction term $g(x,t,u).$
We consider the Cahn-Hilliard equation in one space dimension with scaling a small parameter εand a non-convex potential W. In the limit \espilon \to 0, under the assumption that the initial data are energetically well-prepared, we show the convergence to a Stefan problem. The proof is based on variational methods and exploits the gradient flow structure of the Cahn-Hilliard equation.
The goal of this paper is to solve a long standing open problem, namely, the asymptotic development of order $2$ by $Γ$-convergence of the mass-constrained Cahn-Hilliard functional. This is achieved by introducing a novel rearrangement technique, which works without Dirichlet boundary conditions.
We establish sufficient conditions for a function on the torus to be equal to its Steiner symmetrization and apply the result to volume-constrained minimizers of the Cahn-Hilliard energy. We also show how two-point rearrangements can be used to establish symmetry for the Cahn-Hilliard model. In two dimensions, the Bonnesen inequality can then be applied to quantitatively estimate the sphericity of superlevel sets.
In this paper, the existence of weak solutions of a convective Cahn-Hilliard equation with degenerate mobility is studied. We first define a notion of weak solutions and establish a regularized problems. The existence of such solutions is obtained by considered the limits of the regularized problems.
The existence of an inertial manifold for the 3D Cahn-Hilliard equation with periodic boundary conditions is verified using the proper extension of the so-called spatial averaging principle introduced by G. Sell and J. Mallet-Paret. Moreover, the extra regularity of this manifold is also obtained.
The Cahn-Hilliard equation is a fundamental model that describes the phase separation process in multi-component mixtures. It has been successfully extended to many different contexts in several scientific fields. In this survey article, we briefly review the derivation, structure as well as some analytical issues for the Cahn-Hilliard equation and its variants. Our focus will be placed on the well-posedness and long-time behavior of the Cahn-Hilliard equation in the classical setting and recent progresses on the dynamic boundary conditions accounting for non-trivial boundary effects.
The paper is devoted to the classification of entire solutions to the Cahn-Hilliard equation $-Δu = u-u^3-δ$ in $\R^N$, with particular interest in those solutions whose nodal set is either bounded or contained in a cylinder. The aim is to prove either radial or cylindrical symmetry, under suitable hypothesis.
There has been recently an important interest in deriving rigorously the Cahn-Hilliard equation from the nonlocal equation, also called aggregation equation. So far, only non-degenerate mobilities were treated. Since we are motivated by models for the biomechanics of living tissues, it is useful to include degenerate motilities. In this framework, we present a new method to show the convergence of the nonlocal to the local degenerate Cahn-Hilliard equation. The method includes the use of nonlocal Poincaré and compactness inequalities.
This paper examines the well-posedness of the Stefan problem with a dynamic boundary condition. To show the existence of the weak solution, the original problem is approximated by a limit of an equation and dynamic boundary condition of Cahn-Hilliard type. By using this Cahn-Hilliard approach, it becomes clear that the state of the mushy region of the Stefan problem is characterized by an asymptotic limit of the fourth-order system, which has a double-well structure. This fact also raises the possibility of the numerical application of the Cahn-Hilliard system to the degenerate parabolic equat…
The Cahn-Hilliard and viscous Cahn-Hilliard equations with singular and possibly nonsmooth potentials and dynamic boundary condition are considered and some well-posedness and regularity results are proved. Key words: Cahn-Hilliard equation, dynamic boundary conditions, phase separation, irregular potentials, well-posedness.
This paper studies two well-known models for two-phase fluid flow at constant temperature, the isothermal Navier-Stokes-Allen-Cahn and the isothermal Navier-Stokes-Cahn-Hilliard equations, both of which consist of equations for the (total) fluid density rho, the (mass-averaged)velocity u and the concentration (of one of the phases,) c. Assuming in either case that both phases are incompressible with different densities, each of the models is shown to reduce to a system of evolution equations in rho and u alone. In the case of the Navier-Stokes-Allen-Cahn model, this reduced system is the class…
Using the Maslowski and Seidler method, the existence of invariant measure for 2-dimensional stochastic Cahn-Hilliard-Navier-Stokes equations with multiplicative noise is proved in state space $L_x^2\times H^1$, working with the weak topology. Also, the existence of global pathwise solution is investigated using the stochastic compactness argument.
In this paper the study of a nonlocal second order Cahn-Hilliard-type singularly perturbed family of functions is undertaken. The kernels considered include those leading to Gagliardo fractional seminorms for gradients. Using Gamma convergence the integral representation of the limit energy is characterized leading to an anisotropic surface energy on interfaces separating different phases.
Cahn-Hilliard-Navier-Stokes system describes the evolution of two isothermal, incompressible, immiscible fluids in a bounded domain. In this work, we consider the stationary nonlocal Cahn-Hilliard-Navier-Stokes system in two and three dimensions with singular potential. We prove the existence of a weak solution for the system using pseudo-monotonicity arguments and Browder's theorem. Further we establish the uniqueness and regularity results for the weak solution of the stationary nonlocal Cahn-Hilliard-Navier-Stokes system for constant mobility parameter and viscosity. Finally, in two dimensi…
Phase field models recently gained a lot of interest in the context of tumour growth models. Typically Darcy-type flow models are coupled to Cahn-Hilliard equations. However, often Stokes or Brinkman flows are more appropriate flow models. We introduce and mathematically analyse a new Cahn-Hilliard-Brinkman model for tumour growth allowing for chemotaxis. Outflow boundary conditions are considered in order not to influence tumour growth by artificial boundary conditions. Existence of global-in-time weak solutions is shown in a very general setting.
We investigate a non-isothermal viscous relaxation of some nonlocal Cahn-Hilliard equations. This perturbation problem generates a family of solution operators, exhibiting dissipation and conservation. The solution operators admit a family of compact global attractors that are bounded in a more regular phase-space.
We provide a rigorous mathematical framework to establish the limit of a nonlocal model of cell-cell adhesion system to a local model. When the parameter of the nonlocality goes to 0, the system tends to a Cahn-Hilliard system with degenerate mobility and cross interaction forces. Our analysis relies on a priori estimates and compactness properties.
We study initial boundary value problems for the convective Cahn-Hilliard equation $\Dt u +\px^4u +u\px u+\px^2(|u|^pu)=0$. It is well-known that without the convective term, the solutions of this equation may blow up in finite time for any $p>0$. In contrast to that, we show that the presence of the convective term $u\px u$ in the Cahn-Hilliard equation prevents blow up at least for $0<p<\frac49$. We also show that the blowing up solutions still exist if $p$ is large enough ($p\ge2$). The related equations like Kolmogorov-Sivashinsky-Spiegel equation, sixth order convective Cahn-Hilliard equa…
The well-posedness for a system of partial differential equations and dynamic boundary conditions is discussed. This system is a sort of transmission problem between the dynamics in the bulk $Ω$ and on the boundary $Γ$. The Poisson equation for the chemical potential, the Allen-Cahn equation for the order parameter in the bulk $Ω$ are considered as auxiliary conditions for solving the Cahn-Hilliard equation on the boundary $Γ$. Recently the well-posedness for the equation and dynamic boundary condition, both of Cahn-Hilliard type, was discussed. Based on this result, the existence of the solut…
We construct rigorously suitable approximate solutions to the Stokes/Cahn-Hilliard system by using the method of matched asymptotics expansions. This is a main step in the proof of convergence given in the first part of this contribution, where the rigorous sharp interface limit of a coupled Stokes/Cahn-Hilliard system in a two dimensional, bounded and smooth domain is shown. As a novelty compared to earlier works, we introduce fractional order terms, which are of significant importance, but share the problematic feature that they may not be uniformly estimated in $ε$ in arbitrarily strong nor…
In this paper, we consider the almost sure well-posedness of the Cauchy problem to the Cahn-Hilliard-Navier-Stokes equation with a randomization initial data on a torus $\mathbb{T}^3$. First, we prove the local existence and uniqueness of solution. Furthermore, we prove the global existence and uniqueness of solution and give the relative probability estimate under the condition of small initial data.
We study the infinite-energy solutions of the Cahn-Hilliard equation in the whole 3D space in uniformly local phase spaces. In particular, we establish the global existence of solutions for the case of regular potentials of arbitrary polynomial growth and for the case of sufficiently strong singular potentials. For these cases, the uniqueness and further regularity of the obtained solutions are proved as well. We discuss also the analogous problems for the case of the so-called Cahn-Hilliard-Oono equation where, in addition, the dissipativity of the associated solution semigroup is established…
In this paper, we continue the study of the hyperbolic relaxation of the Cahn-Hilliard-Oono equation with the sub-quintic non-linearity in the whole space $\R^3$ started in our previous paper and verify that under the natural assumptions on the non-linearity and the external force, the fractal dimension of the associated global attractor in the natural energy space is finite.
This paper deals with a parabolic-elliptic chemotaxis system with nonlinear diffusion. It was proved that there exists a solution of a Cahn-Hilliard system as an approximation of a nonlinear diffusion equation by applying an abstract theory by Colli-Visintin [Comm. Partial Differential Equations 15 (1990), 737-756] for a doubly nonlinear evolution inclusion with some bounded monotone operator and subdifferential operator of a proper lower semicontinuous convex function (cf. Colli-Fukao [J. Math. Anal. Appl. 429 (2015), 1190-1213]). Moreover, Colli-Fukao [J. Differential Equations 260 (2016), 6…
In this paper, we study the sharp interface limit for solutions of the Cahn-Hilliard equation with disparate mobilities. This means that the mobility function degenerates in one of the two energetically favorable configurations, suppressing the diffusion in that phase. First, we construct suitable weak solutions to this Cahn-Hilliard equation. Second, we prove precompactness of these solutions under natural assumptions on the initial data. Third, under an additional energy convergence assumption, we show that the sharp interface limit is a distributional solution to the Hele-Shaw flow with opt…
In the present work, we address a class of Cahn-Hilliard equations characterized by a singular diffusion term. The problem is a simplified version with constant mobility of the Cahn-Hilliard-de Gennes model of phase separation in binary, incompressible, isothermal mixtures of polymer molecules. It is proved that, for any final time T, the problem admits a unique energy type weak solution, defined over (0,T). For any s > 0 such solution is classical in the sense of belonging to a suitable Hoelder class over (s,T), and enjoys the property of being separated from the singular values corresponding…
