Cahn-Hilliard Equation Research Articles

Averaging principles for multiscale stochastic Cahn–Hilliard system

In this paper, we will establish averaging principles for the multiscale stochastic Cahn–Hilliard system. The stochastic averaging principle is a powerful tool for studying qualitative analysis of stochastic dynamical systems with different time-scales. Under suitable conditions, two kinds of averaging principle (the autonomous case and the nonautonomous case) are proved, and as a consequence, the multiscale system can be reduced to a single stochastic Cahn–Hilliard equation (averaged equation) with a modified coefficient, the slow component of multiscale stochastic Cahn–Hilliard system towards to the solution of the averaged equation in moment (the autonomous case) and in probability (the nonautonomous case).

Pressure jump and radial stationary solutions of the degenerate Cahn–Hilliard equation

Pressure jump and radial stationary solutions of the degenerate Cahn–Hilliard equation

Improved estimates for the sharp interface limit of the stochastic Cahn–Hilliard equation with space-time white noise

We study the sharp interface limit of the stochastic Cahn–Hilliard equation with cubic double-well potential and additive space-time white noise \varepsilon^{\sigma}\dot{W} , where \varepsilon>0 is an interfacial width parameter. We prove that, for a sufficiently large scaling constant \sigma >0 , the stochastic Cahn–Hilliard equation converges to the deterministic Mullins–Sekerka/Hele-Shaw problem for \varepsilon \rightarrow 0 . The convergence is shown in suitable fractional Sobolev norms as well as in the L^{p} -norm for p\in (2, 4] in spatial dimension d=2,3 . This generalizes the existing result for the space-time white noise to dimension d=3 and improves the existing results for smooth noise, which were so far limited to p\in (2,\frac{2d+8}{d+2}] in spatial dimension d=2,3 . As a byproduct of the analysis of the stochastic problem with space-time white noise, we identify minimal regularity requirements on the noise which allow convergence to the sharp interface limit in the \mathbb{H}^{1} -norm and also provide improved convergence estimates for the sharp interface limit of the deterministic problem.

Efficient energy stable numerical schemes for Cahn–Hilliard equations with dynamical boundary conditions

In this paper, we propose a unified framework for studying the Cahn–Hilliard equation with two distinct types of dynamic boundary conditions, namely, the Allen–Cahn and Cahn–Hilliard types. Using this unified framework, we develop a linear, second-order, and energy-stable scheme based on the multiple scalar auxiliary variables (MSAV) approach. We design efficient and decoupling algorithms for solving the corresponding linear system in which the unknown variables are intricately coupled both in the bulk and at the boundary. Several numerical experiments are shown to validate the proposed scheme, and to investigate the effect of different dynamical boundary conditions on the dynamics of phase evolution under different scenarios.

Nonreciprocal Pattern Formation of Conserved Fields

In recent years, nonreciprocally coupled systems have received growing attention. Previous work has shown that the interplay of nonreciprocal coupling and Goldstone modes can drive the emergence of temporal order such as traveling waves. We show that these phenomena are generically found in a broad class of pattern-forming systems, including mass-conserving reaction-diffusion systems and viscoelastic active gels. All these systems share a characteristic dispersion relation that acquires a nonzero imaginary part at the edge of the band of unstable modes and exhibit a regime of propagating structures (traveling wave bands or droplets). We show that models for these systems can be mapped to a common “normal form” that can be seen as a spatially extended generalization of the FitzHugh-Nagumo model, providing a unifying dynamical-systems perspective. We show that the minimal nonreciprocal Cahn-Hilliard equations exhibit a surprisingly rich set of behaviors, including interrupted coarsening of traveling waves without selection of a preferred wavelength and transversal undulations of wave fronts in two dimensions. We show that the emergence of traveling waves and their speed are precisely predicted from the dispersion relation at interfaces far away from the homogeneous steady state. Our work, thus, generalizes previously studied nonreciprocal phase transitions and shows that interfaces are the relevant collective excitations governing the rich dynamical patterns of conserved fields. Published by the American Physical Society 2024

Integrating factor‐based time integrators for the Cahn–Hilliard equation

In this paper, we propose integrating factor (IF)‐based methods for solving the Cahn–Hilliard (CH) equation, which was originally proposed to describe the phase separation phenomenon for binary melted alloy below the critical temperature, and since then, it has appeared in many fields. Being a nonlinear equation, it is of great importance to develop efficient numerical methods for investigating the solution behavior of the CH equation. We present a few important linear and nonlinear schemes for the CH equation and obtain error estimates for those schemes. The integrators that we present here are of first and second order in time. We show numerical results to illustrate our theoretical findings.

Derivation and analysis of a nonlocal Hele–Shaw–Cahn–Hilliard system for flow in thin heterogeneous layers

We derive, through the deterministic homogenization theory in thin domains, a new model consisting of Hele–Shaw equation with memory coupled with the convective Cahn–Hilliard equation. The obtained system, which models in particular tumor growth, is then analyzed and we prove its well-posedness in dimension 2. To achieve our goal, we develop and use the new concept of sigma-convergence in thin heterogeneous media, and we prove some regularity results for the upscaled model.

On the rate of convergence of Yosida approximation for the nonlocal Cahn–Hilliard equation

Abstract It is well-known that one can construct solutions to the nonlocal Cahn–Hilliard equation with singular potentials via Yosida approximation with parameter $\lambda \to 0$. The usual method is based on compactness arguments and does not provide any rate of convergence. Here, we fill the gap and we obtain an explicit convergence rate $\sqrt{\lambda }$. The proof is based on the theory of maximal monotone operators and an observation that the nonlocal operator is of Hilbert–Schmidt type. Our estimate can provide convergence result for the Galerkin methods where the parameter $\lambda $ could be linked to the discretization parameters, yielding appropriate error estimates.

On the convergence of linear and nonlinear Parareal methods for the Cahn–Hilliard equation

This paper introduces, analyses, and implements efficient time parallel methods for solving the Cahn–Hilliard (CH) equation. Efficient numerical methods for the CH equation are crucial due to its wide range of applications. In particular, simulating the CH equation often requires long computational times to obtain the solution during the phase coarsening stage. Therefore, there is a need to accelerate the computations using parallel methods in the time dimension. We propose linear and nonlinear Parareal methods for the CH equation, depending on the choice of the fine approximation. The effectiveness of our approach is demonstrated through numerical experiments.

Asymptotic stability of the fourth-order $\phi^{4}$ kink for general perturbations in the energy space

The fourth-order \phi^{4} model extends the classical \phi^{4} model of quantum field theory to the fourth-order case, but sharing the same kink solution. It is also the dispersive counterpart of the well-known parabolic Cahn–Hilliard equation. Mathematically speaking, the kink is characterized by a fourth-order nonnegative linear operator with a simple kernel at the origin but no spectral gap. In this paper, we consider the kink of this theory, and prove orbital and asymptotic stability for any perturbation in the energy space.

An efficient data assimilation based unconditionally stable scheme for Cahn–Hilliard equation

An efficient data assimilation based unconditionally stable scheme for Cahn–Hilliard equation

An immersed boundary-phase field fluid-surfactant model with moving contact lines on curved substrates

In this study, we propose an immersed boundary-phase field fluid-surfactant model that incorporates moving contact lines on complex geometries. To accurately capture the interface, we introduce an additional phase field variable to represent the solid phase, which remains fixed throughout the process. The modified Cahn–Hilliard equations are utilized to describe the interface. In order to address stiffness issues, a stabilization technique is incorporated into the calculations. To prevent streamlines from penetrating into solid obstacles, a boundary condition-enforced immersed boundary method is applied at the solid boundaries. The desired contact angle is ensured using the characteristic moving contact line method. All variables are treated independently in the computations, and the numerical results demonstrate the accuracy and excellent performance of our model on curved substrates with surfactants.

On the Doubly Non-local Hele-Shaw–Cahn–Hilliard System: Derivation and 2D Well-Posedness

Starting from a classic non-local (in space) Cahn–Hilliard–Stokes model for two-phase flow in a thin heterogeneous fluid domain, we rigorously derive by mathematical homogenization a new effective mixture model consisting of a coupling of a non-local (in time) Hele-Shaw equation with a non-local (in space) Cahn–Hilliard equation. We then analyse the resulting model and prove its well-posedness. A key to the analysis is the new concept of sigma-convergence in thin heterogeneous domains allowing to pass to the homogenization limit with respect to the heterogeneities and the domain thickness simultaneously.

On the separation property and the global attractor for the nonlocal Cahn-Hilliard equation in three dimensions

We consider the nonlocal Cahn-Hilliard equation with constant mobility and singular potential in three dimensional bounded and smooth domains. This model describes phase separation in binary fluid mixtures. Given any global solution (whose existence and uniqueness are already known), we prove the so-called instantaneous and uniform separation property: any global solution with initial finite energy is globally confined (in the L∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^\\infty $$\\end{document} metric) in the interval [-1+δ,1-δ]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$[-1+\\delta ,1-\\delta ]$$\\end{document} on the time interval [τ,∞)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$[\ au ,\\infty )$$\\end{document} for any τ>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ au >0$$\\end{document}, where δ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\delta $$\\end{document} only depends on the norms of the initial datum, τ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ au $$\\end{document} and the parameters of the system. We then exploit such result to improve the regularity of the global attractor for the dynamical system associated to the problem.

Performance Benchmark of Cahn–Hilliard Equation Solver with Implementation of Semi-implicit Fourier Spectral Method

Performance Benchmark of Cahn–Hilliard Equation Solver with Implementation of Semi-implicit Fourier Spectral Method

Global existence of a pair of coupled Cahn-Hilliard equations with nondegenerate mobility and logarithmic potential

We conducted a mathematical investigation on a system of interconnected Cahn-Hilliard equations featuring a logarithmic potential, nondegenerate mobility, and homogeneous Neumann boundary conditions. This system emerges from a model depicting the phase separation of a binary liquid mixture in a thin film. Assuming certain conditions on the initial data, we successfully established the existence, uniqueness, and stability estimates for the weak solution. Our approach involved initially replacing the logarithmic potential with a smooth counterpart, resulting in the regularization of the original problem (Q) into a regularized problem (Qε). Utilizing the Faedo-Galerkin method and compactness arguments, we demonstrated the existence and uniqueness of a solution for (Qε). Subsequently, by letting ε approach zero, we attained the existence of a solution for the original problem (Q). Additionally, we addressed higher regularity aspects of the weak solutions for both (Q) and (Qε). Employing the standard regularity theory for elliptic problems and introducing additional assumptions regarding the domain's boundary and the initial data, we established that the weak solutions belong to higher-order Sobolev spaces.

Dynamical evolution of spinodal decomposition in holographic superfluids

We study the nonlinear dynamical evolution of spinodal decomposition in a first-order superfluid phase transition using a simple holographic model in the probe limit. We first confirm the linear stability analysis based on quasinormal modes and verify the existence of a critical length scale related to a gradient instability — negative speed of sound squared — of the superfluid sound mode, which is a consequence of a negative thermodynamic charge susceptibility. We present a comparison between our case and the standard Cahn-Hilliard equation for spinodal instability, in which a critical length scale can be also derived based on a diffusive instability. We then perform several numerical tests which include the nonlinear time evolution directly from an unstable state and fast quenches from a stable to an unstable state in the spinodal region. Our numerical results provide a real time description of spinodal decomposition and phase separation in one and two spatial dimensions. We reveal the existence of four different stages in the dynamical evolution, and characterize their main properties. Finally, we investigate the strength of dynamical heterogeneity using the spatial variance of the local chemical potential and we correlate the latter to other features of the dynamical evolution.

Assessment of morphological similarities for the conservative Allen–Cahn and Cahn–Hilliard equations

Phase separations occur in various natural environments, such as materials sciences and biochemistry, and have been scientifically modeled in various fields. We often need to distinguish between two different phase separation phenomena induced by the conservative Allen–Cahn (CAC) and Cahn–Hilliard (CH) equations, which are used to describe natural phenomena. Both phase-field equations share the same properties, including phase-coarsening, mass conservation, and energy dissipation. These similarities make it difficult to determine which equation is responsible for the observed dynamics. However, there are morphological differences between the evolutionary behaviors of the two dynamics, despite their similarities.This work investigates how the dynamics can be characterized by the CAC and CH equations and proposes evaluation criteria to distinguish their features. We examine phase-coarsening in light of the morphological patterns of the phases. Specifically, we train a Convolutional Neural Network (CNN) with morphological patterns generated by the CAC and CH equations. We use Alexnet as one of the CNNs to classify the patterns. In addition, we introduce other geometric properties to characterize the phase-separation dynamics. For a comprehensive understanding of these dynamics, we propose geometrical evaluations such as a distance measure by inner product, area, arclength and anatomical complexity, as measures of similarity metrics. Furthermore, we test and investigate geometrical similarity metrics to discriminate between the CAC and CH equations through numerical experiments. The numerical evaluations of the similarity metrics show that the morphological difference between these two CAC and CH dynamics appears to be large in the middle stages of evolution, whereas the distinction is vague in the early and late stages of evolution. In addition, the geometric aspects of morphological difference and similarity are also confirmed numerically.

Phase-field lattice Boltzmann model with singular mobility for quasi-incompressible two-phase flows.

In this paper, a lattice Boltzmann for quasi-incompressible two-phase flows is proposed based on the Cahn-Hilliard phase-field theory, which can be viewed as an improved model of a previous one [Yang and Guo, Phys. Rev. E 93, 043303 (2016)2470-004510.1103/PhysRevE.93.043303]. The model is composed of two LBE's, one for the Cahn-Hilliard equation(CHE) with a singular mobility, and the other for the quasi-incompressible Navier-Stokes equations(qINSE). Particularly, the LBE for the CHE uses an equilibrium distribution function containing a free parameter associated with the gradient of chemical potential, such that the variable (and even zero) mobility can be handled. In addition, the LBE for the qINSE uses an equilibrium distribution function containing another free parameter associated with the local shear rate, such that the large viscosity ratio problems can be handled. Several tests are first carried out to test the capability of the proposed LBE for the CHE in capturing phase interface, and the results demonstrate that the proposed model outperforms the original LBE model in terms of accuracy and stability. Furthermore, by coupling the hydrodynamic equations, the tests of double-stationary droplets and droplets falling problems indicate that the proposed model can reduce numerical dissipation and produce physically acceptable results at large time scales. The results of droplets falling and phase separation of binary fluid problems show that the present model can handle two-phase flows with large viscosity ratio up to the magnitude of 10^{4}.

The critical variational setting for stochastic evolution equations

In this paper we introduce the critical variational setting for parabolic stochastic evolution equations of quasi- or semi-linear type. Our results improve many of the abstract results in the classical variational setting. In particular, we are able to replace the usual weak or local monotonicity condition by a more flexible local Lipschitz condition. Moreover, the usual growth conditions on the multiplicative noise are weakened considerably. Our new setting provides general conditions under which local and global existence and uniqueness hold. In addition, we prove continuous dependence on the initial data. We show that many classical SPDEs, which could not be covered by the classical variational setting, do fit in the critical variational setting. In particular, this is the case for the Cahn–Hilliard equation, tamed Navier–Stokes equations, and Allen–Cahn equation.