Time-fractional Cahn-Hilliard equation: Well-posedness, degeneracy, and numerical solutions

Barbara Wohlmuth; Mabel L. Rajendran; Marvin Fritz

RECORD

Time-fractional Cahn-Hilliard equation: Well-posedness, degeneracy, and numerical solutions

Marvin Fritz, Mabel L. Rajendran, Barbara Wohlmuth

Analysis of PDEs · Submitted 7 Apr 2021

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim The work concludes with numerical simulations and a sensitivity analysis showing the influence of the fractional power.

how it is stated · asserted 1.00 read the record →

Comment (0) Read on arXiv ↗

Cite ▾

arXiv ID2104.03096 (version 1)

Pith IDDG3VUWBU, Pith's permanent record of this paper and its review

CitationBarbara Wohlmuth, Mabel L. Rajendran, Marvin Fritz. "Time-fractional Cahn-Hilliard equation: Well-posedness, degeneracy, and numerical solutions." arXiv:2104.03096 (2021). https://pith.science/paper/DG3VUWBU

Claim paper

The compiled record

The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Open the compiled record

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 6,185 input tokens.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.33 · human participants 0.01 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.06 · compares with prior results 0.08 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

In this paper, we derive the time-fractional Cahn-Hilliard equation from continuum mixture theory with a modification of Fick's law of diffusion. This model describes the process of phase separation with nonlocal memory effects. We analyze the existence, uniqueness, and regularity of weak solutions of the time-fractional Cahn-Hilliard equation. In this regard, we consider degenerating mobility functions and free energies of Landau, Flory--Huggins and double-obstacle type. We apply the Faedo-Galerkin method to the system, derive energy estimates, and use compactness theorems to pass to the limit in the discrete form. In order to compensate for the missing chain rule of fractional derivatives, we prove a fractional chain inequality for semiconvex functions. The work concludes with numerical simulations and a sensitivity analysis showing the influence of the fractional power. Here, we consider a convolution quadrature scheme for the time-fractional component, and use a mixed finite element method for the space discretization.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	In this paper, we derive the time-fractional Cahn-Hilliard equation from continuum mixture theory with a modification of Fick's law of diffusion.	contribution 0.85	0.96	0.05	asserted 1.00
s1	This model describes the process of phase separation with nonlocal memory effects.	background 0.39	0.35	0.05	asserted 1.00
s2	We analyze the existence, uniqueness, and regularity of weak solutions of the time-fractional Cahn-Hilliard equation.	contribution 0.98	0.95	0.18	asserted 0.95
s3	In this regard, we consider degenerating mobility functions and free energies of Landau, Flory--Huggins and double-obstacle type.	contribution 0.83	0.80	0.03	asserted 0.96
s4	We apply the Faedo-Galerkin method to the system, derive energy estimates, and use compactness theorems to pass to the limit in the discrete form.	method 0.99	0.96	0.06	asserted 1.00
s5	In order to compensate for the missing chain rule of fractional derivatives, we prove a fractional chain inequality for semiconvex functions.	contribution 0.72	0.96	0.06	asserted 1.00
s6	The work concludes with numerical simulations and a sensitivity analysis showing the influence of the fractional power.	contribution 0.65	0.95	0.35	asserted 1.00
s7	Here, we consider a convolution quadrature scheme for the time-fractional component, and use a mixed finite element method for the space discretization.	method 1.00	0.92	0.04	asserted 0.99

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper

Original abstract

In this paper, we derive the time-fractional Cahn-Hilliard equation from continuum mixture theory with a modification of Fick's law of diffusion. This model describes the process of phase separation with nonlocal memory effects. We analyze the existence, uniqueness, and regularity of weak solutions of the time-fractional Cahn-Hilliard equation. In this regard, we consider degenerating mobility functions and free energies of Landau, Flory--Huggins and double-obstacle type. We apply the Faedo-Galerkin method to the system, derive energy estimates, and use compactness theorems to pass to the limit in the discrete form. In order to compensate for the missing chain rule of fractional derivatives, we prove a fractional chain inequality for semiconvex functions. The work concludes with numerical simulations and a sensitivity analysis showing the influence of the fractional power. Here, we consider a convolution quadrature scheme for the time-fractional component, and use a mixed finite element method for the space discretization.

Human review

How Pith checks papers → · Full audit of this paper →

How the literature uses it

2 arXiv papers cite this in 5 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 3 sentences without a confident label

Cites it as background 2 papers

Some of their time-fractional counterparts have been investigated in the literature, e.g., the time-fractional gradient flows of type Allen–Cahn [ref] , Cahn–Hilliard [ref] , Keller–Segel [ref] , and Fokker–Planck [ref] type.

Equivalence between a time-fractional and an integer-order gradient flow: The memory effect reflected in the energy · Introduction · 0.88
The most common numerical methods used for FDEs (see [ref] for overviews) have historically been convolution quadrature [ref] , finite differences [ref] , and finite elements [ref] .

A contour method for time-fractional PDEs and an application to fractional viscoelastic beam equations · Introduction · 0.99

What it cites · 3 sentences without a confident label

Uses the method or tool of 1 paper

We employ the same technique as in [ref] , and approximate and extend the mobility function $m ∈W^1,∞(-1,1)$ with $m(x)>0$ for all $x∈(-1,1)$ and $m(±1)=0$ by a strictly positive function $m_δ$ in the following way: [equation] where $δ∈(0,1)$.

The stochastic Cahn-Hilliard equation with degenerate mobility and logarithmic potential · Degenerating mobility · 0.98

Cites as background 1 paper

Exact solutions have been studied in [ref] and numerical simulations were shown and compared to the time-fractional Allen–Cahn equation, see [ref] .

Solving time-fractional differential equation via rational approximation · Introduction · 0.91

Discussion

Know something about this paper? Leave a comment, or highlight any passage above to comment on it.