---
OA_place: publisher
OA_type: hybrid
_id: '19783'
abstract:
- lang: eng
text: We consider a local Cahn–Hilliard‐type model for tumor growth as well as a
nonlocal model where, compared to the local system, the Laplacian in the equation
for the chemical potential is replaced by a nonlocal operator. The latter is defined
as a convolution integral with suitable kernels parametrized by a small parameter.
For sufficiently smooth bounded domains in three dimensions, we prove convergence
of weak solutions of the nonlocal model toward strong solutions of the local model
together with convergence rates with respect to the small parameter. The proof
is done via a Gronwall‐type argument and a convergence result with rates for the
nonlocal integral operator toward the Laplacian due to Abels and Hurm.
acknowledgement: C. Hurm was partially supported by the Graduiertenkolleg 2339 IntComSin
of the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)–Project-ID
321821685. M. Moser has received funding from the European Research Council (ERC)
under the European Union's Horizon 2020 research and innovation programme (grant
agreement No 948819). The support is gratefully acknowledged. Finally, we thank
Daniel Böhme and Jonas Stange for careful proofreading. Open Access funding enabled
and organized by Projekt DEAL.
article_number: e70003
article_processing_charge: Yes (via OA deal)
article_type: original
arxiv: 1
author:
- first_name: Christoph
full_name: Hurm, Christoph
last_name: Hurm
- first_name: Maximilian
full_name: Moser, Maximilian
id: a60047a9-da77-11eb-85b4-c4dc385ebb8c
last_name: Moser
citation:
ama: Hurm C, Moser M. Nonlocal‐to‐local convergence for a Cahn–Hilliard tumor growth
model. GAMM-Mitteilungen. 2025;48(2). doi:10.1002/gamm.70003
apa: Hurm, C., & Moser, M. (2025). Nonlocal‐to‐local convergence for a Cahn–Hilliard
tumor growth model. GAMM-Mitteilungen. Wiley. https://doi.org/10.1002/gamm.70003
chicago: Hurm, Christoph, and Maximilian Moser. “Nonlocal‐to‐local Convergence for
a Cahn–Hilliard Tumor Growth Model.” GAMM-Mitteilungen. Wiley, 2025. https://doi.org/10.1002/gamm.70003.
ieee: C. Hurm and M. Moser, “Nonlocal‐to‐local convergence for a Cahn–Hilliard tumor
growth model,” GAMM-Mitteilungen, vol. 48, no. 2. Wiley, 2025.
ista: Hurm C, Moser M. 2025. Nonlocal‐to‐local convergence for a Cahn–Hilliard tumor
growth model. GAMM-Mitteilungen. 48(2), e70003.
mla: Hurm, Christoph, and Maximilian Moser. “Nonlocal‐to‐local Convergence for a
Cahn–Hilliard Tumor Growth Model.” GAMM-Mitteilungen, vol. 48, no. 2, e70003,
Wiley, 2025, doi:10.1002/gamm.70003.
short: C. Hurm, M. Moser, GAMM-Mitteilungen 48 (2025).
date_created: 2025-06-03T08:58:01Z
date_published: 2025-06-01T00:00:00Z
date_updated: 2025-06-03T09:14:17Z
day: '01'
ddc:
- '510'
department:
- _id: JuFi
doi: 10.1002/gamm.70003
ec_funded: 1
external_id:
arxiv:
- '2402.13790'
file:
- access_level: open_access
checksum: 6bac9d3e566b68519ae80ac8b0f41f20
content_type: application/pdf
creator: dernst
date_created: 2025-06-03T09:12:22Z
date_updated: 2025-06-03T09:12:22Z
file_id: '19786'
file_name: 2025_GAMM_Hurm.pdf
file_size: 513741
relation: main_file
success: 1
file_date_updated: 2025-06-03T09:12:22Z
has_accepted_license: '1'
intvolume: ' 48'
issue: '2'
language:
- iso: eng
license: https://creativecommons.org/licenses/by/4.0/
month: '06'
oa: 1
oa_version: Published Version
project:
- _id: 0aa76401-070f-11eb-9043-b5bb049fa26d
call_identifier: H2020
grant_number: '948819'
name: Bridging Scales in Random Materials
publication: GAMM-Mitteilungen
publication_identifier:
eissn:
- 1522-2608
issn:
- 0936-7195
publication_status: published
publisher: Wiley
quality_controlled: '1'
scopus_import: '1'
status: public
title: Nonlocal‐to‐local convergence for a Cahn–Hilliard tumor growth model
tmp:
image: /images/cc_by.png
legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
short: CC BY (4.0)
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 48
year: '2025'
...
---
_id: '17887'
abstract:
- lang: eng
text: We show convergence of the Navier-Stokes/Allen-Cahn system to a classical
sharp interface model for the two-phase flow of two viscous incompressible fluids
with same viscosities in a smooth bounded domain in two and three space dimensions
as long as a smooth solution of the limit system exists. Moreover, we obtain error
estimates with the aid of a relative entropy method. Our results hold provided
that the mobility mε>0 in the Allen-Cahn equation tends to zero in a subcritical
way, i.e., mε=m0εβ for some β∈(0,2) and m0>0 . The proof proceeds by showing
via a relative entropy argument that the solution to the Navier-Stokes/Allen-Cahn
system remains close to the solution of a perturbed version of the two-phase flow
problem, augmented by an extra mean curvature flow term mεHΓt in the interface
motion. In a second step, it is easy to see that the solution to the perturbed
problem is close to the original two-phase flow.
acknowledgement: "J. Fischer and M. Moser have received funding from the European
Research Council (ERC) under the European Union’s Horizon 2020 research and innovation
programme (grant agreement No 948819).\r\nOpen Access funding enabled and organized
by Projekt DEAL."
article_number: '77'
article_processing_charge: Yes (via OA deal)
article_type: original
arxiv: 1
author:
- first_name: Helmut
full_name: Abels, Helmut
last_name: Abels
- first_name: Julian L
full_name: Fischer, Julian L
id: 2C12A0B0-F248-11E8-B48F-1D18A9856A87
last_name: Fischer
orcid: 0000-0002-0479-558X
- first_name: Maximilian
full_name: Moser, Maximilian
id: a60047a9-da77-11eb-85b4-c4dc385ebb8c
last_name: Moser
citation:
ama: Abels H, Fischer JL, Moser M. Approximation of classical two-phase flows of
viscous incompressible fluids by a Navier–Stokes/Allen–Cahn system. Archive
for Rational Mechanics and Analysis. 2024;248(5). doi:10.1007/s00205-024-02020-9
apa: Abels, H., Fischer, J. L., & Moser, M. (2024). Approximation of classical
two-phase flows of viscous incompressible fluids by a Navier–Stokes/Allen–Cahn
system. Archive for Rational Mechanics and Analysis. Springer Nature. https://doi.org/10.1007/s00205-024-02020-9
chicago: Abels, Helmut, Julian L Fischer, and Maximilian Moser. “Approximation of
Classical Two-Phase Flows of Viscous Incompressible Fluids by a Navier–Stokes/Allen–Cahn
System.” Archive for Rational Mechanics and Analysis. Springer Nature,
2024. https://doi.org/10.1007/s00205-024-02020-9.
ieee: H. Abels, J. L. Fischer, and M. Moser, “Approximation of classical two-phase
flows of viscous incompressible fluids by a Navier–Stokes/Allen–Cahn system,”
Archive for Rational Mechanics and Analysis, vol. 248, no. 5. Springer
Nature, 2024.
ista: Abels H, Fischer JL, Moser M. 2024. Approximation of classical two-phase flows
of viscous incompressible fluids by a Navier–Stokes/Allen–Cahn system. Archive
for Rational Mechanics and Analysis. 248(5), 77.
mla: Abels, Helmut, et al. “Approximation of Classical Two-Phase Flows of Viscous
Incompressible Fluids by a Navier–Stokes/Allen–Cahn System.” Archive for Rational
Mechanics and Analysis, vol. 248, no. 5, 77, Springer Nature, 2024, doi:10.1007/s00205-024-02020-9.
short: H. Abels, J.L. Fischer, M. Moser, Archive for Rational Mechanics and Analysis
248 (2024).
date_created: 2024-09-08T22:01:10Z
date_published: 2024-09-03T00:00:00Z
date_updated: 2025-09-08T09:11:41Z
day: '03'
ddc:
- '510'
department:
- _id: JuFi
doi: 10.1007/s00205-024-02020-9
ec_funded: 1
external_id:
arxiv:
- '2311.02997'
isi:
- '001305530600001'
pmid:
- '39239088'
file:
- access_level: open_access
checksum: 98493a05b84e4513b6394dfad4851ddf
content_type: application/pdf
creator: dernst
date_created: 2024-09-09T08:43:32Z
date_updated: 2024-09-09T08:43:32Z
file_id: '17938'
file_name: 2024_ArchiveRatAnalysis_Abels.pdf
file_size: 811131
relation: main_file
success: 1
file_date_updated: 2024-09-09T08:43:32Z
has_accepted_license: '1'
intvolume: ' 248'
isi: 1
issue: '5'
language:
- iso: eng
month: '09'
oa: 1
oa_version: Published Version
pmid: 1
project:
- _id: 0aa76401-070f-11eb-9043-b5bb049fa26d
call_identifier: H2020
grant_number: '948819'
name: Bridging Scales in Random Materials
publication: Archive for Rational Mechanics and Analysis
publication_identifier:
eissn:
- 1432-0673
issn:
- 0003-9527
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: Approximation of classical two-phase flows of viscous incompressible fluids
by a Navier–Stokes/Allen–Cahn system
tmp:
image: /images/cc_by.png
legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
short: CC BY (4.0)
type: journal_article
user_id: 317138e5-6ab7-11ef-aa6d-ffef3953e345
volume: 248
year: '2024'
...
---
_id: '15334'
abstract:
- lang: eng
text: We consider the sharp interface limit of a Navier-Stokes/Allen Cahn equation
in a bounded smooth domain in two space dimensions, in the case of vanishing mobility
mε=ε√, where the small parameter ε>0 related to the thickness of the diffuse interface
is sent to zero. For well-prepared initial data and sufficiently small times,
we rigorously prove convergence to the classical two-phase Navier-Stokes system
with surface tension. The idea of the proof is to use asymptotic expansions to
construct an approximate solution and to estimate the difference of the exact
and approximate solutions with a spectral estimate for the (at the approximate
solution) linearized Allen-Cahn operator. In the calculations we use a fractional
order ansatz and new ansatz terms in higher orders leading to a suitable ε-scaled
and coupled model problem. Moreover, we apply the novel idea of introducing ε-dependent
coordinates.
acknowledgement: "Open Access funding enabled and organized by Projekt DEAL.\r\nM.
Fei was partially supported by NSF of China under Grant No. 12271004 and Anhui Provincial
Funding Project under Grant Nos. gxbjZD2022009 and 2308085J10. Moreover, M. Moser
has received funding from the European Research Council (ERC) under the European
Union’s Horizon 2020 research and innovation programme (Grant Agreement No 948819)."
article_number: '94'
article_processing_charge: Yes (via OA deal)
article_type: original
arxiv: 1
author:
- first_name: Helmut
full_name: Abels, Helmut
last_name: Abels
- first_name: Mingwen
full_name: Fei, Mingwen
last_name: Fei
- first_name: Maximilian
full_name: Moser, Maximilian
id: a60047a9-da77-11eb-85b4-c4dc385ebb8c
last_name: Moser
citation:
ama: Abels H, Fei M, Moser M. Sharp interface limit for a Navier–Stokes/Allen–Cahn
system in the case of a vanishing mobility. Calculus of Variations and Partial
Differential Equations. 2024;63(4). doi:10.1007/s00526-024-02715-7
apa: Abels, H., Fei, M., & Moser, M. (2024). Sharp interface limit for a Navier–Stokes/Allen–Cahn
system in the case of a vanishing mobility. Calculus of Variations and Partial
Differential Equations. Springer Nature. https://doi.org/10.1007/s00526-024-02715-7
chicago: Abels, Helmut, Mingwen Fei, and Maximilian Moser. “Sharp Interface Limit
for a Navier–Stokes/Allen–Cahn System in the Case of a Vanishing Mobility.” Calculus
of Variations and Partial Differential Equations. Springer Nature, 2024. https://doi.org/10.1007/s00526-024-02715-7.
ieee: H. Abels, M. Fei, and M. Moser, “Sharp interface limit for a Navier–Stokes/Allen–Cahn
system in the case of a vanishing mobility,” Calculus of Variations and Partial
Differential Equations, vol. 63, no. 4. Springer Nature, 2024.
ista: Abels H, Fei M, Moser M. 2024. Sharp interface limit for a Navier–Stokes/Allen–Cahn
system in the case of a vanishing mobility. Calculus of Variations and Partial
Differential Equations. 63(4), 94.
mla: Abels, Helmut, et al. “Sharp Interface Limit for a Navier–Stokes/Allen–Cahn
System in the Case of a Vanishing Mobility.” Calculus of Variations and Partial
Differential Equations, vol. 63, no. 4, 94, Springer Nature, 2024, doi:10.1007/s00526-024-02715-7.
short: H. Abels, M. Fei, M. Moser, Calculus of Variations and Partial Differential
Equations 63 (2024).
date_created: 2024-04-21T22:00:52Z
date_published: 2024-05-01T00:00:00Z
date_updated: 2025-09-04T13:45:40Z
day: '01'
ddc:
- '510'
department:
- _id: JuFi
doi: 10.1007/s00526-024-02715-7
ec_funded: 1
external_id:
arxiv:
- '2304.12096'
isi:
- '001199418100002'
file:
- access_level: open_access
checksum: b1095fad4cae596f52cc616a973bdde2
content_type: application/pdf
creator: dernst
date_created: 2024-04-23T07:30:48Z
date_updated: 2024-04-23T07:30:48Z
file_id: '15343'
file_name: 2024_CalculusEquations_Abels.pdf
file_size: 975186
relation: main_file
success: 1
file_date_updated: 2024-04-23T07:30:48Z
has_accepted_license: '1'
intvolume: ' 63'
isi: 1
issue: '4'
language:
- iso: eng
month: '05'
oa: 1
oa_version: Published Version
project:
- _id: 0aa76401-070f-11eb-9043-b5bb049fa26d
call_identifier: H2020
grant_number: '948819'
name: Bridging Scales in Random Materials
publication: Calculus of Variations and Partial Differential Equations
publication_identifier:
eissn:
- 1432-0835
issn:
- 0944-2669
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: Sharp interface limit for a Navier–Stokes/Allen–Cahn system in the case of
a vanishing mobility
tmp:
image: /images/cc_by.png
legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
short: CC BY (4.0)
type: journal_article
user_id: 317138e5-6ab7-11ef-aa6d-ffef3953e345
volume: 63
year: '2024'
...
---
_id: '14755'
abstract:
- lang: eng
text: We consider the sharp interface limit for the scalar-valued and vector-valued
Allen–Cahn equation with homogeneous Neumann boundary condition in a bounded smooth
domain Ω of arbitrary dimension N ⩾ 2 in the situation when a two-phase diffuse
interface has developed and intersects the boundary ∂ Ω. The limit problem is
mean curvature flow with 90°-contact angle and we show convergence in strong norms
for well-prepared initial data as long as a smooth solution to the limit problem
exists. To this end we assume that the limit problem has a smooth solution on
[ 0 , T ] for some time T > 0. Based on the latter we construct suitable curvilinear
coordinates and set up an asymptotic expansion for the scalar-valued and the vector-valued
Allen–Cahn equation. In order to estimate the difference of the exact and approximate
solutions with a Gronwall-type argument, a spectral estimate for the linearized
Allen–Cahn operator in both cases is required. The latter will be shown in a separate
paper, cf. (Moser (2021)).
acknowledgement: "The author gratefully acknowledges support through DFG, GRK 1692
“Curvature,\r\nCycles and Cohomology” during parts of the work."
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Maximilian
full_name: Moser, Maximilian
id: a60047a9-da77-11eb-85b4-c4dc385ebb8c
last_name: Moser
citation:
ama: 'Moser M. Convergence of the scalar- and vector-valued Allen–Cahn equation
to mean curvature flow with 90°-contact angle in higher dimensions, part I: Convergence
result. Asymptotic Analysis. 2023;131(3-4):297-383. doi:10.3233/asy-221775'
apa: 'Moser, M. (2023). Convergence of the scalar- and vector-valued Allen–Cahn
equation to mean curvature flow with 90°-contact angle in higher dimensions, part
I: Convergence result. Asymptotic Analysis. IOS Press. https://doi.org/10.3233/asy-221775'
chicago: 'Moser, Maximilian. “Convergence of the Scalar- and Vector-Valued Allen–Cahn
Equation to Mean Curvature Flow with 90°-Contact Angle in Higher Dimensions, Part
I: Convergence Result.” Asymptotic Analysis. IOS Press, 2023. https://doi.org/10.3233/asy-221775.'
ieee: 'M. Moser, “Convergence of the scalar- and vector-valued Allen–Cahn equation
to mean curvature flow with 90°-contact angle in higher dimensions, part I: Convergence
result,” Asymptotic Analysis, vol. 131, no. 3–4. IOS Press, pp. 297–383,
2023.'
ista: 'Moser M. 2023. Convergence of the scalar- and vector-valued Allen–Cahn equation
to mean curvature flow with 90°-contact angle in higher dimensions, part I: Convergence
result. Asymptotic Analysis. 131(3–4), 297–383.'
mla: 'Moser, Maximilian. “Convergence of the Scalar- and Vector-Valued Allen–Cahn
Equation to Mean Curvature Flow with 90°-Contact Angle in Higher Dimensions, Part
I: Convergence Result.” Asymptotic Analysis, vol. 131, no. 3–4, IOS Press,
2023, pp. 297–383, doi:10.3233/asy-221775.'
short: M. Moser, Asymptotic Analysis 131 (2023) 297–383.
corr_author: '1'
date_created: 2024-01-08T13:13:28Z
date_published: 2023-02-02T00:00:00Z
date_updated: 2025-09-09T14:14:55Z
day: '02'
department:
- _id: JuFi
doi: 10.3233/asy-221775
external_id:
arxiv:
- '2105.07100'
isi:
- '000927801300001'
intvolume: ' 131'
isi: 1
issue: 3-4
keyword:
- General Mathematics
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://doi.org/10.48550/arXiv.2105.07100
month: '02'
oa: 1
oa_version: Preprint
page: 297-383
publication: Asymptotic Analysis
publication_identifier:
eissn:
- 1875-8576
issn:
- 0921-7134
publication_status: published
publisher: IOS Press
quality_controlled: '1'
scopus_import: '1'
status: public
title: 'Convergence of the scalar- and vector-valued Allen–Cahn equation to mean curvature
flow with 90°-contact angle in higher dimensions, part I: Convergence result'
type: journal_article
user_id: 317138e5-6ab7-11ef-aa6d-ffef3953e345
volume: 131
year: '2023'
...
---
_id: '12079'
abstract:
- lang: eng
text: We extend the recent rigorous convergence result of Abels and Moser (SIAM
J Math Anal 54(1):114–172, 2022. https://doi.org/10.1137/21M1424925) concerning
convergence rates for solutions of the Allen–Cahn equation with a nonlinear Robin
boundary condition towards evolution by mean curvature flow with constant contact
angle. More precisely, in the present work we manage to remove the perturbative
assumption on the contact angle being close to 90∘. We establish under usual double-well
type assumptions on the potential and for a certain class of boundary energy densities
the sub-optimal convergence rate of order ε12 for general contact angles α∈(0,π).
For a very specific form of the boundary energy density, we even obtain from our
methods a sharp convergence rate of order ε; again for general contact angles
α∈(0,π). Our proof deviates from the popular strategy based on rigorous asymptotic
expansions and stability estimates for the linearized Allen–Cahn operator. Instead,
we follow the recent approach by Fischer et al. (SIAM J Math Anal 52(6):6222–6233,
2020. https://doi.org/10.1137/20M1322182), thus relying on a relative entropy
technique. We develop a careful adaptation of their approach in order to encode
the constant contact angle condition. In fact, we perform this task at the level
of the notion of gradient flow calibrations. This concept was recently introduced
in the context of weak-strong uniqueness for multiphase mean curvature flow by
Fischer et al. (arXiv:2003.05478v2).
acknowledgement: "This Project has received funding from the European Research Council
(ERC) under the European Union’s Horizon 2020 research and innovation programme
(Grant Agreement No 948819) , and from the Deutsche Forschungsgemeinschaft (DFG,
German Research Foundation) under Germany’s Excellence Strategy—EXC-2047/1 - 390685813.\r\nOpen
Access funding enabled and organized by Projekt DEAL."
article_number: '201'
article_processing_charge: No
article_type: original
author:
- first_name: Sebastian
full_name: Hensel, Sebastian
id: 4D23B7DA-F248-11E8-B48F-1D18A9856A87
last_name: Hensel
orcid: 0000-0001-7252-8072
- first_name: Maximilian
full_name: Moser, Maximilian
id: a60047a9-da77-11eb-85b4-c4dc385ebb8c
last_name: Moser
citation:
ama: 'Hensel S, Moser M. Convergence rates for the Allen–Cahn equation with boundary
contact energy: The non-perturbative regime. Calculus of Variations and Partial
Differential Equations. 2022;61(6). doi:10.1007/s00526-022-02307-3'
apa: 'Hensel, S., & Moser, M. (2022). Convergence rates for the Allen–Cahn equation
with boundary contact energy: The non-perturbative regime. Calculus of Variations
and Partial Differential Equations. Springer Nature. https://doi.org/10.1007/s00526-022-02307-3'
chicago: 'Hensel, Sebastian, and Maximilian Moser. “Convergence Rates for the Allen–Cahn
Equation with Boundary Contact Energy: The Non-Perturbative Regime.” Calculus
of Variations and Partial Differential Equations. Springer Nature, 2022. https://doi.org/10.1007/s00526-022-02307-3.'
ieee: 'S. Hensel and M. Moser, “Convergence rates for the Allen–Cahn equation with
boundary contact energy: The non-perturbative regime,” Calculus of Variations
and Partial Differential Equations, vol. 61, no. 6. Springer Nature, 2022.'
ista: 'Hensel S, Moser M. 2022. Convergence rates for the Allen–Cahn equation with
boundary contact energy: The non-perturbative regime. Calculus of Variations and
Partial Differential Equations. 61(6), 201.'
mla: 'Hensel, Sebastian, and Maximilian Moser. “Convergence Rates for the Allen–Cahn
Equation with Boundary Contact Energy: The Non-Perturbative Regime.” Calculus
of Variations and Partial Differential Equations, vol. 61, no. 6, 201, Springer
Nature, 2022, doi:10.1007/s00526-022-02307-3.'
short: S. Hensel, M. Moser, Calculus of Variations and Partial Differential Equations
61 (2022).
date_created: 2022-09-11T22:01:54Z
date_published: 2022-08-24T00:00:00Z
date_updated: 2025-04-14T07:53:59Z
day: '24'
ddc:
- '510'
department:
- _id: JuFi
doi: 10.1007/s00526-022-02307-3
ec_funded: 1
external_id:
isi:
- '000844247300008'
file:
- access_level: open_access
checksum: b2da020ce50440080feedabeab5b09c4
content_type: application/pdf
creator: dernst
date_created: 2023-01-20T08:56:01Z
date_updated: 2023-01-20T08:56:01Z
file_id: '12320'
file_name: 2022_Calculus_Hensel.pdf
file_size: 1278493
relation: main_file
success: 1
file_date_updated: 2023-01-20T08:56:01Z
has_accepted_license: '1'
intvolume: ' 61'
isi: 1
issue: '6'
language:
- iso: eng
month: '08'
oa: 1
oa_version: Published Version
project:
- _id: 0aa76401-070f-11eb-9043-b5bb049fa26d
call_identifier: H2020
grant_number: '948819'
name: Bridging Scales in Random Materials
publication: Calculus of Variations and Partial Differential Equations
publication_identifier:
eissn:
- 1432-0835
issn:
- 0944-2669
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: 'Convergence rates for the Allen–Cahn equation with boundary contact energy:
The non-perturbative regime'
tmp:
image: /images/cc_by.png
legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
short: CC BY (4.0)
type: journal_article
user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
volume: 61
year: '2022'
...
---
_id: '12305'
abstract:
- lang: eng
text: This paper is concerned with the sharp interface limit for the Allen--Cahn
equation with a nonlinear Robin boundary condition in a bounded smooth domain
Ω⊂\R2. We assume that a diffuse interface already has developed and that it is
in contact with the boundary ∂Ω. The boundary condition is designed in such a
way that the limit problem is given by the mean curvature flow with constant α-contact
angle. For α close to 90° we prove a local in time convergence result for well-prepared
initial data for times when a smooth solution to the limit problem exists. Based
on the latter we construct a suitable curvilinear coordinate system and carry
out a rigorous asymptotic expansion for the Allen--Cahn equation with the nonlinear
Robin boundary condition. Moreover, we show a spectral estimate for the corresponding
linearized Allen--Cahn operator and with its aid we derive strong norm estimates
for the difference of the exact and approximate solutions using a Gronwall-type
argument.
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Helmut
full_name: Abels, Helmut
last_name: Abels
- first_name: Maximilian
full_name: Moser, Maximilian
id: a60047a9-da77-11eb-85b4-c4dc385ebb8c
last_name: Moser
citation:
ama: Abels H, Moser M. Convergence of the Allen--Cahn equation with a nonlinear
Robin boundary condition to mean curvature flow with contact angle close to 90°.
SIAM Journal on Mathematical Analysis. 2022;54(1):114-172. doi:10.1137/21m1424925
apa: Abels, H., & Moser, M. (2022). Convergence of the Allen--Cahn equation
with a nonlinear Robin boundary condition to mean curvature flow with contact
angle close to 90°. SIAM Journal on Mathematical Analysis. Society for
Industrial and Applied Mathematics. https://doi.org/10.1137/21m1424925
chicago: Abels, Helmut, and Maximilian Moser. “Convergence of the Allen--Cahn Equation
with a Nonlinear Robin Boundary Condition to Mean Curvature Flow with Contact
Angle Close to 90°.” SIAM Journal on Mathematical Analysis. Society for
Industrial and Applied Mathematics, 2022. https://doi.org/10.1137/21m1424925.
ieee: H. Abels and M. Moser, “Convergence of the Allen--Cahn equation with a nonlinear
Robin boundary condition to mean curvature flow with contact angle close to 90°,”
SIAM Journal on Mathematical Analysis, vol. 54, no. 1. Society for Industrial
and Applied Mathematics, pp. 114–172, 2022.
ista: Abels H, Moser M. 2022. Convergence of the Allen--Cahn equation with a nonlinear
Robin boundary condition to mean curvature flow with contact angle close to 90°.
SIAM Journal on Mathematical Analysis. 54(1), 114–172.
mla: Abels, Helmut, and Maximilian Moser. “Convergence of the Allen--Cahn Equation
with a Nonlinear Robin Boundary Condition to Mean Curvature Flow with Contact
Angle Close to 90°.” SIAM Journal on Mathematical Analysis, vol. 54, no.
1, Society for Industrial and Applied Mathematics, 2022, pp. 114–72, doi:10.1137/21m1424925.
short: H. Abels, M. Moser, SIAM Journal on Mathematical Analysis 54 (2022) 114–172.
corr_author: '1'
date_created: 2023-01-16T10:07:00Z
date_published: 2022-01-04T00:00:00Z
date_updated: 2024-10-09T21:03:58Z
day: '04'
department:
- _id: JuFi
doi: 10.1137/21m1424925
external_id:
arxiv:
- '2105.08434'
isi:
- '000762768000004'
intvolume: ' 54'
isi: 1
issue: '1'
keyword:
- Applied Mathematics
- Computational Mathematics
- Analysis
language:
- iso: eng
main_file_link:
- open_access: '1'
url: ' https://doi.org/10.48550/arXiv.2105.08434'
month: '01'
oa: 1
oa_version: Preprint
page: 114-172
publication: SIAM Journal on Mathematical Analysis
publication_identifier:
eissn:
- 1095-7154
issn:
- 0036-1410
publication_status: published
publisher: Society for Industrial and Applied Mathematics
quality_controlled: '1'
scopus_import: '1'
status: public
title: Convergence of the Allen--Cahn equation with a nonlinear Robin boundary condition
to mean curvature flow with contact angle close to 90°
type: journal_article
user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
volume: 54
year: '2022'
...