The Cahn–Hilliard equation with a nonlinear source term

2021 ◽  
Vol 294 ◽  
pp. 88-117
Author(s):  
Alain Miranville
Keyword(s):  
Source Term ◽  
Nonlinear Source ◽  
Hilliard Equation ◽  
Cahn Hilliard Equation

scholarly journals Existence and limit problem for fractional fourth order subdiffusion equation and Cahn-Hilliard equation

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Nguyen Huy Tuan
Keyword(s):  
Mild Solution ◽  
Caputo Derivative ◽  
Source Term ◽  
Fixed Point Theory ◽  
Local Existence ◽  
Point Theory ◽  
Limit Problem ◽  
Heat Semigroup ◽  
Hilliard Equation ◽  
Cahn Hilliard Equation

<p style='text-indent:20px;'>In this paper, we study fractional subdiffusion fourth parabolic equations containing Caputo and Caputo-Fabrizio operators. The main results of the paper are presented in two parts. For the first part with the Caputo derivative, we focus on the global and local well-posedness results. We study the global mild solution for biharmonic heat equation with Caputo derivative in the case of globally Lipschitz source term. A new weighted space is used for this case. We then proceed to give the results about the local existence in the case of locally Lipschitz source term. To overcome the intricacies of the proofs, we applied <inline-formula><tex-math id="M1">\begin{document}$ L^p-L^q $\end{document}</tex-math></inline-formula> estimate for biharmonic heat semigroup, Banach fixed point theory, some estimates for Mittag-Lefler functions and Wright functions, and also Sobolev embeddings. For the second result involving the Cahn-Hilliard equation with the Caputo-Fabrizio operator, we first show the local existence result. In addition, we first provide that the connections of the mild solution between the Cahn-Hilliard equation in the case <inline-formula><tex-math id="M2">\begin{document}$ 0&lt;{\alpha}&lt;1 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M3">\begin{document}$ {\alpha} = 1 $\end{document}</tex-math></inline-formula>. This is the first result of investigating the Cahn-Hilliard equation with this type of derivative. The main key of the proof is based on complex evaluations involving exponential functions, and some embeddings between <inline-formula><tex-math id="M4">\begin{document}$ L^p $\end{document}</tex-math></inline-formula> spaces and Hilbert scales spaces.</p>

scholarly journals Benchmark Problems for the Numerical Discretization of the Cahn–Hilliard Equation with a Source Term

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Sungha Yoon ◽  
Hyun Geun Lee ◽  
Yibao Li ◽  
Chaeyoung Lee ◽  
Jintae Park ◽  
...  
Keyword(s):  
Laplace Operator ◽  
Source Term ◽  
Orientation Dependence ◽  
Benchmark Problems ◽  
Hilliard Equation ◽  
Growth Phenomenon ◽  
Discrete Laplace Operator ◽  
Space Discretization ◽  
Numerical Discretization ◽  
Cahn Hilliard Equation

In this paper, we present benchmark problems for the numerical discretization of the Cahn–Hilliard equation with a source term. If the source term includes an isotropic growth term, then initially circular and spherical shapes should grow with their original shapes. However, there is numerical anisotropic error and this error results in anisotropic evolutions. Therefore, it is essential to use isotropic space discretization in the simulation of growth phenomenon such as tumor growth. To test numerical discretization, we present two benchmark problems: one is the growth of a disk or a sphere and the other is the growth of a rotated ellipse or a rotated ellipsoid. The computational results show that the standard discrete Laplace operator has severe grid orientation dependence. However, the isotropic discrete Laplace operator generates good results.

scholarly journals On the sharp interface limit of a model for phase separation on biological membranes

Analysis ◽  
2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Helmut Abels ◽  
Johannes Kampmann
Keyword(s):  
Lipid Raft ◽  
Cell Membranes ◽  
System Modeling ◽  
Bulk Diffusion ◽  
Type Equation ◽  
Sharp Interface ◽  
Interface Problem ◽  
Hilliard Equation ◽  
System A ◽  
Cahn Hilliard Equation

AbstractWe rigorously prove the convergence of weak solutions to a model for lipid raft formation in cell membranes which was recently proposed in [H. Garcke, J. Kampmann, A. Rätz and M. Röger, A coupled surface-Cahn–Hilliard bulk-diffusion system modeling lipid raft formation in cell membranes, Math. Models Methods Appl. Sci. 26 2016, 6, 1149–1189] to weak (varifold) solutions of the corresponding sharp-interface problem for a suitable subsequence. In the system a Cahn–Hilliard type equation on the boundary of a domain is coupled to a diffusion equation inside the domain. The proof builds on techniques developed in [X. Chen, Global asymptotic limit of solutions of the Cahn–Hilliard equation, J. Differential Geom. 44 1996, 2, 262–311] for the corresponding result for the Cahn–Hilliard equation.

scholarly journals On global dynamics of 2D convective Cahn–Hilliard equation

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Xiaopeng Zhao
Keyword(s):  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Global Dynamics ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Initial Value ◽  
Hilliard Equation ◽  
Long Time ◽  
Cahn Hilliard Equation ◽  
Initial Boundary Value

AbstractIn this paper, we study the long time behavior of solution for the initial-boundary value problem of convective Cahn–Hilliard equation in a 2D case. We show that the equation has a global attractor in $H^{4}(\Omega )$ H 4 ( Ω ) when the initial value belongs to $H^{1}(\Omega )$ H 1 ( Ω ) .

scholarly journals A free–energy stable p–adaptive nodal discontinuous Galerkin for the Cahn–Hilliard equation

2021 ◽  
pp. 110409
Author(s):  
Gerasimos Ntoukas ◽  
Juan Manzanero ◽  
Gonzalo Rubio ◽  
Eusebio Valero ◽  
Esteban Ferrer
Keyword(s):  
Free Energy ◽  
Discontinuous Galerkin ◽  
Hilliard Equation ◽  
Energy Stable ◽  
Cahn Hilliard Equation
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