---
_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. <i>Calculus of Variations and Partial
    Differential Equations</i>. 2024;63(4). doi:<a href="https://doi.org/10.1007/s00526-024-02715-7">10.1007/s00526-024-02715-7</a>
  apa: Abels, H., Fei, M., &#38; Moser, M. (2024). Sharp interface limit for a Navier–Stokes/Allen–Cahn
    system in the case of a vanishing mobility. <i>Calculus of Variations and Partial
    Differential Equations</i>. Springer Nature. <a href="https://doi.org/10.1007/s00526-024-02715-7">https://doi.org/10.1007/s00526-024-02715-7</a>
  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.” <i>Calculus
    of Variations and Partial Differential Equations</i>. Springer Nature, 2024. <a
    href="https://doi.org/10.1007/s00526-024-02715-7">https://doi.org/10.1007/s00526-024-02715-7</a>.
  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,” <i>Calculus of Variations and Partial
    Differential Equations</i>, 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.” <i>Calculus of Variations and Partial
    Differential Equations</i>, vol. 63, no. 4, 94, Springer Nature, 2024, doi:<a
    href="https://doi.org/10.1007/s00526-024-02715-7">10.1007/s00526-024-02715-7</a>.
  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
license: https://creativecommons.org/licenses/by/4.0/
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'
...