---
_id: '9246'
abstract:
- lang: eng
  text: We consider the Fröhlich Hamiltonian in a mean-field limit where many bosonic
    particles weakly couple to the quantized phonon field. For large particle numbers
    and a suitably small coupling, we show that the dynamics of the system is approximately
    described by the Landau–Pekar equations. These describe a Bose–Einstein condensate
    interacting with a classical polarization field, whose dynamics is effected by
    the condensate, i.e., the back-reaction of the phonons that are created by the
    particles during the time evolution is of leading order.
acknowledgement: "Financial support by the European Research Council (ERC) under the\r\nEuropean
  Union’s Horizon 2020 research and innovation programme (Grant Agreement\r\nNo 694227;
  N.L and R.S.), the SNSF Eccellenza Project PCEFP2 181153 (N.L) and the\r\nDeutsche
  Forschungsgemeinschaft (DFG) through the Research TrainingGroup 1838: Spectral\r\nTheory
  and Dynamics of Quantum Systems (D.M.) is gratefully acknowledged. N.L.\r\ngratefully
  acknowledges support from the NCCRSwissMAP and would like to thank Simone\r\nRademacher
  and Benjamin Schlein for interesting discussions about the time-evolution of\r\nthe
  polaron at strong coupling. D.M. thanks Marcel Griesemer and Andreas Wünsch for\r\nextensive
  discussions about the Fröhlich polaron."
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Nikolai K
  full_name: Leopold, Nikolai K
  id: 4BC40BEC-F248-11E8-B48F-1D18A9856A87
  last_name: Leopold
  orcid: 0000-0002-0495-6822
- first_name: David Johannes
  full_name: Mitrouskas, David Johannes
  id: cbddacee-2b11-11eb-a02e-a2e14d04e52d
  last_name: Mitrouskas
- first_name: Robert
  full_name: Seiringer, Robert
  id: 4AFD0470-F248-11E8-B48F-1D18A9856A87
  last_name: Seiringer
  orcid: 0000-0002-6781-0521
citation:
  ama: Leopold NK, Mitrouskas DJ, Seiringer R. Derivation of the Landau–Pekar equations
    in a many-body mean-field limit. <i>Archive for Rational Mechanics and Analysis</i>.
    2021;240:383-417. doi:<a href="https://doi.org/10.1007/s00205-021-01616-9">10.1007/s00205-021-01616-9</a>
  apa: Leopold, N. K., Mitrouskas, D. J., &#38; Seiringer, R. (2021). Derivation of
    the Landau–Pekar equations in a many-body mean-field limit. <i>Archive for Rational
    Mechanics and Analysis</i>. Springer Nature. <a href="https://doi.org/10.1007/s00205-021-01616-9">https://doi.org/10.1007/s00205-021-01616-9</a>
  chicago: Leopold, Nikolai K, David Johannes Mitrouskas, and Robert Seiringer. “Derivation
    of the Landau–Pekar Equations in a Many-Body Mean-Field Limit.” <i>Archive for
    Rational Mechanics and Analysis</i>. Springer Nature, 2021. <a href="https://doi.org/10.1007/s00205-021-01616-9">https://doi.org/10.1007/s00205-021-01616-9</a>.
  ieee: N. K. Leopold, D. J. Mitrouskas, and R. Seiringer, “Derivation of the Landau–Pekar
    equations in a many-body mean-field limit,” <i>Archive for Rational Mechanics
    and Analysis</i>, vol. 240. Springer Nature, pp. 383–417, 2021.
  ista: Leopold NK, Mitrouskas DJ, Seiringer R. 2021. Derivation of the Landau–Pekar
    equations in a many-body mean-field limit. Archive for Rational Mechanics and
    Analysis. 240, 383–417.
  mla: Leopold, Nikolai K., et al. “Derivation of the Landau–Pekar Equations in a
    Many-Body Mean-Field Limit.” <i>Archive for Rational Mechanics and Analysis</i>,
    vol. 240, Springer Nature, 2021, pp. 383–417, doi:<a href="https://doi.org/10.1007/s00205-021-01616-9">10.1007/s00205-021-01616-9</a>.
  short: N.K. Leopold, D.J. Mitrouskas, R. Seiringer, Archive for Rational Mechanics
    and Analysis 240 (2021) 383–417.
date_created: 2021-03-14T23:01:34Z
date_published: 2021-02-26T00:00:00Z
date_updated: 2023-08-07T14:12:27Z
day: '26'
ddc:
- '510'
department:
- _id: RoSe
doi: 10.1007/s00205-021-01616-9
ec_funded: 1
external_id:
  arxiv:
  - '2001.03993'
  isi:
  - '000622226200001'
file:
- access_level: open_access
  checksum: 23449e44dc5132501a5c86e70638800f
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  creator: dernst
  date_created: 2021-03-22T08:31:29Z
  date_updated: 2021-03-22T08:31:29Z
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language:
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oa_version: Published Version
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project:
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  call_identifier: H2020
  grant_number: '694227'
  name: Analysis of quantum many-body systems
publication: Archive for Rational Mechanics and Analysis
publication_identifier:
  eissn:
  - '14320673'
  issn:
  - '00039527'
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: Derivation of the Landau–Pekar equations in a many-body mean-field limit
tmp:
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type: journal_article
user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
volume: 240
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...
---
_id: '7489'
abstract:
- lang: eng
  text: 'In the present work, we consider the evolution of two fluids separated by
    a sharp interface in the presence of surface tension—like, for example, the evolution
    of oil bubbles in water. Our main result is a weak–strong uniqueness principle
    for the corresponding free boundary problem for the incompressible Navier–Stokes
    equation: as long as a strong solution exists, any varifold solution must coincide
    with it. In particular, in the absence of physical singularities, the concept
    of varifold solutions—whose global in time existence has been shown by Abels (Interfaces
    Free Bound 9(1):31–65, 2007) for general initial data—does not introduce a mechanism
    for non-uniqueness. The key ingredient of our approach is the construction of
    a relative entropy functional capable of controlling the interface error. If the
    viscosities of the two fluids do not coincide, even for classical (strong) solutions
    the gradient of the velocity field becomes discontinuous at the interface, introducing
    the need for a careful additional adaption of the relative entropy.'
article_processing_charge: Yes (via OA deal)
article_type: original
author:
- 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: Sebastian
  full_name: Hensel, Sebastian
  id: 4D23B7DA-F248-11E8-B48F-1D18A9856A87
  last_name: Hensel
  orcid: 0000-0001-7252-8072
citation:
  ama: Fischer JL, Hensel S. Weak–strong uniqueness for the Navier–Stokes equation
    for two fluids with surface tension. <i>Archive for Rational Mechanics and Analysis</i>.
    2020;236:967-1087. doi:<a href="https://doi.org/10.1007/s00205-019-01486-2">10.1007/s00205-019-01486-2</a>
  apa: Fischer, J. L., &#38; Hensel, S. (2020). Weak–strong uniqueness for the Navier–Stokes
    equation for two fluids with surface tension. <i>Archive for Rational Mechanics
    and Analysis</i>. Springer Nature. <a href="https://doi.org/10.1007/s00205-019-01486-2">https://doi.org/10.1007/s00205-019-01486-2</a>
  chicago: Fischer, Julian L, and Sebastian Hensel. “Weak–Strong Uniqueness for the
    Navier–Stokes Equation for Two Fluids with Surface Tension.” <i>Archive for Rational
    Mechanics and Analysis</i>. Springer Nature, 2020. <a href="https://doi.org/10.1007/s00205-019-01486-2">https://doi.org/10.1007/s00205-019-01486-2</a>.
  ieee: J. L. Fischer and S. Hensel, “Weak–strong uniqueness for the Navier–Stokes
    equation for two fluids with surface tension,” <i>Archive for Rational Mechanics
    and Analysis</i>, vol. 236. Springer Nature, pp. 967–1087, 2020.
  ista: Fischer JL, Hensel S. 2020. Weak–strong uniqueness for the Navier–Stokes equation
    for two fluids with surface tension. Archive for Rational Mechanics and Analysis.
    236, 967–1087.
  mla: Fischer, Julian L., and Sebastian Hensel. “Weak–Strong Uniqueness for the Navier–Stokes
    Equation for Two Fluids with Surface Tension.” <i>Archive for Rational Mechanics
    and Analysis</i>, vol. 236, Springer Nature, 2020, pp. 967–1087, doi:<a href="https://doi.org/10.1007/s00205-019-01486-2">10.1007/s00205-019-01486-2</a>.
  short: J.L. Fischer, S. Hensel, Archive for Rational Mechanics and Analysis 236
    (2020) 967–1087.
date_created: 2020-02-16T23:00:50Z
date_published: 2020-05-01T00:00:00Z
date_updated: 2023-09-07T13:30:45Z
day: '01'
ddc:
- '530'
- '532'
department:
- _id: JuFi
doi: 10.1007/s00205-019-01486-2
ec_funded: 1
external_id:
  isi:
  - '000511060200001'
file:
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  checksum: f107e21b58f5930876f47144be37cf6c
  content_type: application/pdf
  creator: dernst
  date_created: 2020-11-20T09:14:22Z
  date_updated: 2020-11-20T09:14:22Z
  file_id: '8779'
  file_name: 2020_ArchRatMechAn_Fischer.pdf
  file_size: 1897571
  relation: main_file
  success: 1
file_date_updated: 2020-11-20T09:14:22Z
has_accepted_license: '1'
intvolume: '       236'
isi: 1
language:
- iso: eng
month: '05'
oa: 1
oa_version: Published Version
page: 967-1087
project:
- _id: 2564DBCA-B435-11E9-9278-68D0E5697425
  call_identifier: H2020
  grant_number: '665385'
  name: International IST Doctoral Program
- _id: B67AFEDC-15C9-11EA-A837-991A96BB2854
  name: IST Austria Open Access Fund
publication: Archive for Rational Mechanics and Analysis
publication_identifier:
  eissn:
  - '14320673'
  issn:
  - '00039527'
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
related_material:
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  - id: '10007'
    relation: dissertation_contains
    status: public
scopus_import: '1'
status: public
title: Weak–strong uniqueness for the Navier–Stokes equation for two fluids with surface
  tension
tmp:
  image: /images/cc_by.png
  legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
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  short: CC BY (4.0)
type: journal_article
user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
volume: 236
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...