Nonlocal Cahn Hilliard Equation Research Articles

Nonlocal Cahn–Hilliard Equation with Degenerate Mobility: Incompressible Limit and Convergence to Stationary States

Nonlocal Cahn–Hilliard Equation with Degenerate Mobility: Incompressible Limit and Convergence to Stationary States

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 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.

Regularity results and optimal velocity control of the convective nonlocal Cahn-Hilliard equation in 3D

In this contribution, we study an optimal control problem for the celebrated nonlocal Cahn-Hilliard equation endowed with the singular Flory-Huggins potential in the three-dimensional setting. The control enters the governing state system in a nonlinear fashion in the form of a prescribed solenoidal, that is a divergence-free, vector field, whereas the cost functional to be minimized is of tracking-type. The novelties of the present paper are twofold: in addition to the control application, the intrinsic difficulties of the optimization problem forced us to first establish new regularity results on the nonlocal Cahn-Hilliard equation that were unknown even without the coupling with a velocity field and are therefore of independent interest. This happens to be shown using the recently proved separation property along with ad hoc Hölder regularities and a bootstrap method. For the control problem, the existence of an optimal strategy as well as first-order necessary conditions are then established.

Energy-decreasing second order exponential time differencing Runge–Kutta methods for Nonlocal Cahn–Hilliard equation

Due to the energetic variational structure inherent in the nonlocal phase field model, the energy of the system naturally decreases over time according to the Nonlocal Cahn–Hilliard (NCH) equation. Preserving this crucial property of energy dissipation is highly desirable in numerical solutions. However, most existing numerical methods can only ensure a decrease in some modified form of energy, rather than the original energy itself. In this paper, we propose the second-order exponential time differencing Runge–Kutta (ETD-RK2) methods for solving the NCH equation. We demonstrate that this method has the ability to unconditionally preserve the original energy dissipation. Numerical experiments are included to illustrate the accuracy and energy stability of the ETD-RK2 methods.

Exponential stability and stabilization of a stochastic 2D nonlocal Cahn-Hilliard-Navier-Stokes equations with multiplicative noise

In this article, we study the asymptotic stability of the unique strong solution to a stochastic version of a diffuse interface model which describes the motion of an incompressible isothermal mixture of two immiscible fluids. The model consists of the 2D Navier-Stokes equations coupled with a convective nonlocal Cahn-Hilliard equation. We establish the existence, uniqueness and regularity results for the stationary solution of the stochastic 2D nonlocal Cahn-Hilliard Navier-Stokes equations. Furthermore, we prove that under some conditions on the forcing terms, the strong solution converges exponentially in the mean square and almost surely exponentially to the stationary solution. Finally, we also prove a result related to the stabilization of the stochastic 2D nonlocal Cahn-Hilliard-Navier-Stokes equations.

Fully discrete, decoupled and energy-stable Fourier-Spectral numerical scheme for the nonlocal Cahn–Hilliard equation coupled with Navier–Stokes/Darcy flow regime of two-phase incompressible flows

In this paper, we introduce fully discrete Fourier-Spectral numerical scheme to solve the nonlocal Cahn–Hilliard equation coupled with Navier–Stokes/Darcy equations, which represent a phase-field model for two-phase incompressible flow in either the free flow regime or a Hele-Shaw cell. The proposed scheme achieves full decoupling while maintaining linearity and energy stability through a combination of the Scalar Auxiliary Variable (SAV) method, which discretizes the nonlinear potential, and the “Zero-Energy-Contribution” (ZEC) method, which handles the coupled nonlinear advective/surface tension terms. The efficiency of this scheme is attributed to its linear decoupling structure and the fact that it requires only a few elliptic equations with constant coefficients to be solved at each time step. We rigorously establish the scheme’s unconditional energy stability. Further, some numerical simulations are provided in both 2D and 3D to show its effectiveness, including its accuracy, stability, and efficiency.

Cahn–Hilliard equations on random walk spaces

In this paper, we study a nonlocal Cahn–Hilliard equation (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 [Formula: see text] of the Cahn–Hilliard Equation. We also show that the Cahn–Hilliard equation is the gradient flow of the Ginzburg–Landau free energy functional on an appropriate Hilbert space. We finally study the asymptotic behavior of the solutions.

Double stabilizations and convergence analysis of a second-order linear numerical scheme for the nonlocal Cahn-Hilliard equation

Double stabilizations and convergence analysis of a second-order linear numerical scheme for the nonlocal Cahn-Hilliard equation

From Vlasov Equation to Degenerate Nonlocal Cahn-Hilliard Equation

We provide a rigorous mathematical framework to establish the hydrodynamic limit of the Vlasov model introduced in Takata and Noguchi (J. Stat. Phys. 172:880-903, 2018) by Noguchi and Takata in order to describe phase transition of fluids by kinetic equations. We prove that, when the scale parameter tends to 0, this model converges to a nonlocal Cahn-Hilliard equation with degenerate mobility. For our analysis, we introduce apropriate forms of the short and long range potentials which allow us to derive Helmhotlz free energy estimates. Several compactness properties follow from the energy, the energy dissipation and kinetic averaging lemmas. In particular we prove a new weak compactness bound on the flux.

Arbitrarily high-order energy stable s-stage RK-IEQ scheme for the nonlocal Cahn–Hilliard equation

In this paper, for solving the nonlocal-type Cahn–Hilliard equation, we propose an arbitrary high-order energy stable s-stage RK-IEQ scheme, which is based on the combination of the Invariant Energy Quantization (IEQ) approach and the s-stage Runge–Kutta method. We prove that the proposed scheme is unconditionally energy stable and carry out several numerical examples in both 2D and 3D to demonstrate the accuracy and stability of the scheme.

Weak solution of a stochastic 3D nonlocal Cahn–Hilliard–Navier–Stokes systems with shear-dependent viscosity

We consider the stochastic nonlocal Cahn–Hilliard–Navier–Stokes system with shear-dependent viscosity on a bounded domain , d = 2, 3, driven by a multiplicative noise of Lévy and Gaussian types. The velocity u is governed by a Navier–Stokes system with a shear-dependent viscosity controlled by a power p>2. This system is nonlinearly coupled through the Korteweg force with a convective nonlocal Cahn–Hilliard equation for the order parameter φ. The existence of a global weak martingale solution is proved. In the 2D case, we prove the pathwise uniqueness of the weak solution, when .

The existence of Non-blow-up phenomenon for a generalized nonlocal Cahn–Hilliard equation with degenerate mobility

The existence of Non-blow-up phenomenon for a generalized nonlocal Cahn–Hilliard equation with degenerate mobility

Stabilization parameter analysis of a second-order linear numerical scheme for the nonlocal Cahn–Hilliard equation

AbstractA second-order accurate (in time) and linear numerical scheme is proposed and analyzed for the nonlocal Cahn–Hilliard equation. The backward differentiation formula is used as the temporal discretization, while an explicit extrapolation is applied to the nonlinear term and the concave expansive term. In addition, an $O (\varDelta {t}^2)$ artificial regularization term, in the form of $A \varDelta _N (\phi ^{n+1} - 2 \phi ^n + \phi ^{n-1})$, is added for the sake of numerical stability. The resulting constant-coefficient linear scheme brings great numerical convenience; however, its theoretical analysis turns out to be very challenging, due to the lack of higher-order diffusion in the nonlocal model. In fact, a rough energy stability analysis can be derived, where an assumption on the $\ell ^\infty $ bound of the numerical solution is required. To recover such an $\ell ^\infty $ bound, an optimal rate convergence analysis has to be conducted, which combines a high-order consistency analysis for the numerical system and the stability estimate for the error function. We adopt a novel test function for the error equation, so that a higher-order temporal truncation error is derived to match the accuracy for discretizing the temporal derivative. Under the view that the numerical solution is actually a small perturbation of the exact solution, a uniform $\ell ^\infty $ bound of the numerical solution can be obtained, by resorting to the error estimate under a moderate constraint of the time step size. Therefore, the result of the energy stability is restated with a new assumption on the stabilization parameter $A$. Some numerical experiments are carried out to display the behavior of the proposed second-order scheme, including the convergence tests and long-time coarsening dynamics.

Convergence of a nonlocal to a local diffuse interface model for two-phase flow with unmatched densities

<p style='text-indent:20px;'>We prove convergence of suitable subsequences of weak solutions of a diffuse interface model for the two-phase flow of incompressible fluids with different densities with a nonlocal Cahn-Hilliard equation to weak solutions of the corresponding system with a standard "local" Cahn-Hilliard equation. The analysis is done in the case of a sufficiently smooth bounded domain with no-slip boundary condition for the velocity and Neumann boundary conditions for the Cahn-Hilliard equation. The proof is based on the corresponding result in the case of a single Cahn-Hilliard equation and compactness arguments used in the proof of existence of weak solutions for the diffuse interface model.</p>

Nonlocal Cahn–Hilliard–Hele–Shaw Systems with Singular Potential and Degenerate Mobility

We study a Cahn-Hilliard-Hele-Shaw (or Cahn-Hilliard-Darcy) system for an incompressible mixture of two fluids. The relative concentration difference $\varphi$ is governed by a convective nonlocal Cahn-Hilliard equation with degenerate mobility and logarithmic potential. The volume averaged fluid velocity $\mathbf{u}$ obeys a Darcy's law depending on the so-called Korteweg force $\mu\nabla \varphi$, where $\mu$ is the nonlocal chemical potential. In addition, the kinematic viscosity $\eta$ may depend on $\varphi$. We establish first the existence of a global weak solution which satisfies the energy identity. Then we prove the existence of a strong solution. Further regularity results on the pressure and on $\mathbf{u}$ are also obtained. Weak-strong uniqueness is demonstrated in the two dimensional case. In the three-dimensional case, uniqueness of weak solutions holds if $\eta$ is constant. Otherwise, weak-strong uniqueness is shown by assuming that the pressure of the strong solution is $\alpha$-H\"{o}lder continuous in space for $\alpha\in (1/5,1)$.

Finite element solution of nonlocal Cahn–Hilliard equations with feedback control time step size adaptivity

AbstractIn this study, we evaluate the performance of feedback control‐based time step adaptivity schemes for the nonlocal Cahn–Hilliard equation derived from the Ohta–Kawasaki free energy functional. The temporal adaptivity scheme is recast under the linear feedback control theory equipped with an error estimation that extrapolates the solution obtained from an energy‐stable, fully implicit time marching scheme. We test three time step controllers with different properties: a simple Integral controller, a complete proportional‐integral‐derivative controller, and the PC11 predictive controller. We assess the performance of the adaptive schemes for the nonlocal Cahn–Hilliard equation in terms of the number of time steps required for the complete simulation and the computational effort measured by the required number of nonlinear and linear solver iterations. We also present numerical evidence of mass conservation and free energy decay for simulations with the three different time step controllers. The PC11 predictive controller is the best in all three‐dimensional test cases.

On a class of non-local phase-field models for tumor growth with possibly singular potentials, chemotaxis, and active transport

This paper provides a unified mathematical analysis of a family of non-local diffuse interface models for tumor growth describing evolutions driven by long-range interactions. These integro-partial differential equations model cell-to-cell adhesion by a non-local term and may be seen as non-local variants of the corresponding local model proposed by Garcke et al (2016). The model in consideration couples a non-local Cahn–Hilliard equation for the tumor phase variable with a reaction–diffusion equation for the nutrient concentration, and takes into account also significant mechanisms such as chemotaxis and active transport. The system depends on two relaxation parameters: a viscosity coefficient and parabolic-regularization coefficient on the chemical potential. The first part of the paper is devoted to the analysis of the system with both regularizations. Here, a rich spectrum of results is presented. Weak well-posedness is first addressed, also including singular potentials. Then, under suitable conditions, existence of strong solutions enjoying the separation property is proved. This allows also to obtain a refined stability estimate with respect to the data, including both chemotaxis and active transport. The second part of the paper is devoted to the study of the asymptotic behavior of the system as the relaxation parameters vanish. The asymptotics are analyzed when the parameters approach zero both separately and jointly, and exact error estimates are obtained. As a by-product, well-posedness of the corresponding limit systems is established.

Regularity results for the nonlocal Cahn-Hilliard equation with singular potential and degenerate mobility

We consider the nonlocal Cahn-Hilliard equation with singular potential and degenerate mobility in a bounded domain Ω⊂Rd, d≤3. We first prove the existence of maximal strong solutions in weighted (in time) Lp spaces. Then we establish further regularity properties of the solution through maximal regularity theory. Finally, we revisit the separation property in an appendix.

Strong well-posedness and inverse identification problem of a non-local phase field tumour model with degenerate mobilities

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, 3–24). The model consists of a non-local Cahn–Hilliard equation with degenerate mobility and singular potential for the phase field variable, coupled to a reaction–diffusion equation for the concentration of a nutrient. We prove the existence of strong solutions to the model and establish some high-order continuous dependence estimates, even in the presence of concentration-dependent mobilities for the nutrient variable in two spatial dimensions. Then, we apply the new regularity results to study an inverse problem identifying the initial tumour distribution from measurements at the terminal time. Formulating the Tikhonov regularised inverse problem as a constrained minimisation problem, we establish the existence of minimisers and derive first-order necessary optimality conditions.