---
_id: '13043'
abstract:
- lang: eng
text: "We derive a weak-strong uniqueness principle for BV solutions to multiphase
mean curvature flow of triple line clusters in three dimensions. Our proof is
based on the explicit construction\r\nof a gradient flow calibration in the sense
of the recent work of Fischer et al. (2020) for any such\r\ncluster. This extends
the two-dimensional construction to the three-dimensional case of surfaces\r\nmeeting
along triple junctions."
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.
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Sebastian
full_name: Hensel, Sebastian
id: 4D23B7DA-F248-11E8-B48F-1D18A9856A87
last_name: Hensel
orcid: 0000-0001-7252-8072
- first_name: Tim
full_name: Laux, Tim
last_name: Laux
citation:
ama: Hensel S, Laux T. Weak-strong uniqueness for the mean curvature flow of double
bubbles. Interfaces and Free Boundaries. 2023;25(1):37-107. doi:10.4171/IFB/484
apa: Hensel, S., & Laux, T. (2023). Weak-strong uniqueness for the mean curvature
flow of double bubbles. Interfaces and Free Boundaries. EMS Press. https://doi.org/10.4171/IFB/484
chicago: Hensel, Sebastian, and Tim Laux. “Weak-Strong Uniqueness for the Mean Curvature
Flow of Double Bubbles.” Interfaces and Free Boundaries. EMS Press, 2023.
https://doi.org/10.4171/IFB/484.
ieee: S. Hensel and T. Laux, “Weak-strong uniqueness for the mean curvature flow
of double bubbles,” Interfaces and Free Boundaries, vol. 25, no. 1. EMS
Press, pp. 37–107, 2023.
ista: Hensel S, Laux T. 2023. Weak-strong uniqueness for the mean curvature flow
of double bubbles. Interfaces and Free Boundaries. 25(1), 37–107.
mla: Hensel, Sebastian, and Tim Laux. “Weak-Strong Uniqueness for the Mean Curvature
Flow of Double Bubbles.” Interfaces and Free Boundaries, vol. 25, no. 1,
EMS Press, 2023, pp. 37–107, doi:10.4171/IFB/484.
short: S. Hensel, T. Laux, Interfaces and Free Boundaries 25 (2023) 37–107.
date_created: 2023-05-21T22:01:06Z
date_published: 2023-04-20T00:00:00Z
date_updated: 2023-08-01T14:43:29Z
day: '20'
ddc:
- '510'
department:
- _id: JuFi
doi: 10.4171/IFB/484
ec_funded: 1
external_id:
arxiv:
- '2108.01733'
isi:
- '000975817300002'
file:
- access_level: open_access
checksum: 622422484810441e48f613e968c7e7a4
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creator: dernst
date_created: 2023-05-22T07:24:13Z
date_updated: 2023-05-22T07:24:13Z
file_id: '13045'
file_name: 2023_Interfaces_Hensel.pdf
file_size: 867876
relation: main_file
success: 1
file_date_updated: 2023-05-22T07:24:13Z
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intvolume: ' 25'
isi: 1
issue: '1'
language:
- iso: eng
license: https://creativecommons.org/licenses/by/4.0/
month: '04'
oa: 1
oa_version: Published Version
page: 37-107
project:
- _id: 0aa76401-070f-11eb-9043-b5bb049fa26d
call_identifier: H2020
grant_number: '948819'
name: Bridging Scales in Random Materials
publication: Interfaces and Free Boundaries
publication_identifier:
eissn:
- 1463-9971
issn:
- 1463-9963
publication_status: published
publisher: EMS Press
quality_controlled: '1'
related_material:
record:
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relation: earlier_version
status: public
scopus_import: '1'
status: public
title: Weak-strong uniqueness for the mean curvature flow of double bubbles
tmp:
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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: 25
year: '2023'
...
---
_id: '11842'
abstract:
- lang: eng
text: We consider the flow of two viscous and incompressible fluids within a bounded
domain modeled by means of a two-phase Navier–Stokes system. The two fluids are
assumed to be immiscible, meaning that they are separated by an interface. With
respect to the motion of the interface, we consider pure transport by the fluid
flow. Along the boundary of the domain, a complete slip boundary condition for
the fluid velocities and a constant ninety degree contact angle condition for
the interface are assumed. In the present work, we devise for the resulting evolution
problem a suitable weak solution concept based on the framework of varifolds and
establish as the main result a weak-strong uniqueness principle in 2D. The proof
is based on a relative entropy argument and requires a non-trivial further development
of ideas from the recent work of Fischer and the first author (Arch. Ration. Mech.
Anal. 236, 2020) to incorporate the contact angle condition. To focus on the effects
of the necessarily singular geometry of the evolving fluid domains, we work for
simplicity in the regime of same viscosities for the two fluids.
acknowledgement: The authors warmly thank their former resp. current PhD advisor Julian
Fischer for the suggestion of this problem and for valuable initial discussions
on the subjects of this paper. 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.
article_number: '93'
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Sebastian
full_name: Hensel, Sebastian
id: 4D23B7DA-F248-11E8-B48F-1D18A9856A87
last_name: Hensel
orcid: 0000-0001-7252-8072
- first_name: Alice
full_name: Marveggio, Alice
id: 25647992-AA84-11E9-9D75-8427E6697425
last_name: Marveggio
citation:
ama: Hensel S, Marveggio A. Weak-strong uniqueness for the Navier–Stokes equation
for two fluids with ninety degree contact angle and same viscosities. Journal
of Mathematical Fluid Mechanics. 2022;24(3). doi:10.1007/s00021-022-00722-2
apa: Hensel, S., & Marveggio, A. (2022). Weak-strong uniqueness for the Navier–Stokes
equation for two fluids with ninety degree contact angle and same viscosities.
Journal of Mathematical Fluid Mechanics. Springer Nature. https://doi.org/10.1007/s00021-022-00722-2
chicago: Hensel, Sebastian, and Alice Marveggio. “Weak-Strong Uniqueness for the
Navier–Stokes Equation for Two Fluids with Ninety Degree Contact Angle and Same
Viscosities.” Journal of Mathematical Fluid Mechanics. Springer Nature,
2022. https://doi.org/10.1007/s00021-022-00722-2.
ieee: S. Hensel and A. Marveggio, “Weak-strong uniqueness for the Navier–Stokes
equation for two fluids with ninety degree contact angle and same viscosities,”
Journal of Mathematical Fluid Mechanics, vol. 24, no. 3. Springer Nature,
2022.
ista: Hensel S, Marveggio A. 2022. Weak-strong uniqueness for the Navier–Stokes
equation for two fluids with ninety degree contact angle and same viscosities.
Journal of Mathematical Fluid Mechanics. 24(3), 93.
mla: Hensel, Sebastian, and Alice Marveggio. “Weak-Strong Uniqueness for the Navier–Stokes
Equation for Two Fluids with Ninety Degree Contact Angle and Same Viscosities.”
Journal of Mathematical Fluid Mechanics, vol. 24, no. 3, 93, Springer Nature,
2022, doi:10.1007/s00021-022-00722-2.
short: S. Hensel, A. Marveggio, Journal of Mathematical Fluid Mechanics 24 (2022).
date_created: 2022-08-14T22:01:45Z
date_published: 2022-08-01T00:00:00Z
date_updated: 2023-11-30T13:25:02Z
day: '01'
ddc:
- '510'
department:
- _id: JuFi
doi: 10.1007/s00021-022-00722-2
ec_funded: 1
external_id:
arxiv:
- '2112.11154'
isi:
- '000834834300001'
file:
- access_level: open_access
checksum: 75c5f286300e6f0539cf57b4dba108d5
content_type: application/pdf
creator: cchlebak
date_created: 2022-08-16T06:55:22Z
date_updated: 2022-08-16T06:55:22Z
file_id: '11848'
file_name: 2022_JMathFluidMech_Hensel.pdf
file_size: 2045570
relation: main_file
success: 1
file_date_updated: 2022-08-16T06:55:22Z
has_accepted_license: '1'
intvolume: ' 24'
isi: 1
issue: '3'
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: Journal of Mathematical Fluid Mechanics
publication_identifier:
eissn:
- 1422-6952
issn:
- 1422-6928
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
related_material:
record:
- id: '14587'
relation: dissertation_contains
status: public
scopus_import: '1'
status: public
title: Weak-strong uniqueness for the Navier–Stokes equation for two fluids with ninety
degree contact angle and same viscosities
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: 24
year: '2022'
...
---
_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: 2023-08-03T13:48:30Z
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: '9307'
abstract:
- lang: eng
text: We establish finite time extinction with probability one for weak solutions
of the Cauchy–Dirichlet problem for the 1D stochastic porous medium equation with
Stratonovich transport noise and compactly supported smooth initial datum. Heuristically,
this is expected to hold because Brownian motion has average spread rate O(t12)
whereas the support of solutions to the deterministic PME grows only with rate
O(t1m+1). The rigorous proof relies on a contraction principle up to time-dependent
shift for Wong–Zakai type approximations, the transformation to a deterministic
PME with two copies of a Brownian path as the lateral boundary, and techniques
from the theory of viscosity solutions.
acknowledgement: This project has received funding from the European Union’s Horizon
2020 research and innovation programme under the Marie Skłodowska-Curie Grant Agreement
No. 665385 . I am very grateful to M. Gerencsér and J. Maas for proposing this problem
as well as helpful discussions. Special thanks go to F. Cornalba for suggesting
the additional κ-truncation in Proposition 5. I am also indebted to an anonymous
referee for pointing out a gap in a previous version of the proof of Lemma 9 (concerning
the treatment of the noise term). The issue is resolved in this version.
article_processing_charge: Yes (via OA deal)
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
citation:
ama: 'Hensel S. Finite time extinction for the 1D stochastic porous medium equation
with transport noise. Stochastics and Partial Differential Equations: Analysis
and Computations. 2021;9:892–939. doi:10.1007/s40072-021-00188-9'
apa: 'Hensel, S. (2021). Finite time extinction for the 1D stochastic porous medium
equation with transport noise. Stochastics and Partial Differential Equations:
Analysis and Computations. Springer Nature. https://doi.org/10.1007/s40072-021-00188-9'
chicago: 'Hensel, Sebastian. “Finite Time Extinction for the 1D Stochastic Porous
Medium Equation with Transport Noise.” Stochastics and Partial Differential
Equations: Analysis and Computations. Springer Nature, 2021. https://doi.org/10.1007/s40072-021-00188-9.'
ieee: 'S. Hensel, “Finite time extinction for the 1D stochastic porous medium equation
with transport noise,” Stochastics and Partial Differential Equations: Analysis
and Computations, vol. 9. Springer Nature, pp. 892–939, 2021.'
ista: 'Hensel S. 2021. Finite time extinction for the 1D stochastic porous medium
equation with transport noise. Stochastics and Partial Differential Equations:
Analysis and Computations. 9, 892–939.'
mla: 'Hensel, Sebastian. “Finite Time Extinction for the 1D Stochastic Porous Medium
Equation with Transport Noise.” Stochastics and Partial Differential Equations:
Analysis and Computations, vol. 9, Springer Nature, 2021, pp. 892–939, doi:10.1007/s40072-021-00188-9.'
short: 'S. Hensel, Stochastics and Partial Differential Equations: Analysis and
Computations 9 (2021) 892–939.'
date_created: 2021-04-04T22:01:21Z
date_published: 2021-03-21T00:00:00Z
date_updated: 2023-08-07T14:31:59Z
day: '21'
ddc:
- '510'
department:
- _id: JuFi
doi: 10.1007/s40072-021-00188-9
ec_funded: 1
external_id:
isi:
- '000631001700001'
file:
- access_level: open_access
checksum: 6529b609c9209861720ffa4685111bc6
content_type: application/pdf
creator: dernst
date_created: 2021-04-06T09:31:28Z
date_updated: 2021-04-06T09:31:28Z
file_id: '9309'
file_name: 2021_StochPartDiffEquation_Hensel.pdf
file_size: 727005
relation: main_file
success: 1
file_date_updated: 2021-04-06T09:31:28Z
has_accepted_license: '1'
intvolume: ' 9'
isi: 1
language:
- iso: eng
month: '03'
oa: 1
oa_version: Published Version
page: 892–939
project:
- _id: 2564DBCA-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '665385'
name: International IST Doctoral Program
publication: 'Stochastics and Partial Differential Equations: Analysis and Computations'
publication_identifier:
eissn:
- 2194-041X
issn:
- 2194-0401
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: Finite time extinction for the 1D stochastic porous medium equation with transport
noise
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: 9
year: '2021'
...
---
_id: '10007'
abstract:
- lang: eng
text: The present thesis is concerned with the derivation of weak-strong uniqueness
principles for curvature driven interface evolution problems not satisfying a
comparison principle. The specific examples being treated are two-phase Navier-Stokes
flow with surface tension, modeling the evolution of two incompressible, viscous
and immiscible fluids separated by a sharp interface, and multiphase mean curvature
flow, which serves as an idealized model for the motion of grain boundaries in
an annealing polycrystalline material. Our main results - obtained in joint works
with Julian Fischer, Tim Laux and Theresa M. Simon - state that prior to the formation
of geometric singularities due to topology changes, the weak solution concept
of Abels (Interfaces Free Bound. 9, 2007) to two-phase Navier-Stokes flow with
surface tension and the weak solution concept of Laux and Otto (Calc. Var. Partial
Differential Equations 55, 2016) to multiphase mean curvature flow (for networks
in R^2 or double bubbles in R^3) represents the unique solution to these interface
evolution problems within the class of classical solutions, respectively. To the
best of the author's knowledge, for interface evolution problems not admitting
a geometric comparison principle the derivation of a weak-strong uniqueness principle
represented an open problem, so that the works contained in the present thesis
constitute the first positive results in this direction. The key ingredient of
our approach consists of the introduction of a novel concept of relative entropies
for a class of curvature driven interface evolution problems, for which the associated
energy contains an interfacial contribution being proportional to the surface
area of the evolving (network of) interface(s). The interfacial part of the relative
entropy gives sufficient control on the interface error between a weak and a classical
solution, and its time evolution can be computed, at least in principle, for any
energy dissipating weak solution concept. A resulting stability estimate for the
relative entropy essentially entails the above mentioned weak-strong uniqueness
principles. The present thesis contains a detailed introduction to our relative
entropy approach, which in particular highlights potential applications to other
problems in curvature driven interface evolution not treated in this thesis.
alternative_title:
- ISTA Thesis
article_processing_charge: No
author:
- first_name: Sebastian
full_name: Hensel, Sebastian
id: 4D23B7DA-F248-11E8-B48F-1D18A9856A87
last_name: Hensel
orcid: 0000-0001-7252-8072
citation:
ama: 'Hensel S. Curvature driven interface evolution: Uniqueness properties of weak
solution concepts. 2021. doi:10.15479/at:ista:10007'
apa: 'Hensel, S. (2021). Curvature driven interface evolution: Uniqueness properties
of weak solution concepts. Institute of Science and Technology Austria. https://doi.org/10.15479/at:ista:10007'
chicago: 'Hensel, Sebastian. “Curvature Driven Interface Evolution: Uniqueness Properties
of Weak Solution Concepts.” Institute of Science and Technology Austria, 2021.
https://doi.org/10.15479/at:ista:10007.'
ieee: 'S. Hensel, “Curvature driven interface evolution: Uniqueness properties of
weak solution concepts,” Institute of Science and Technology Austria, 2021.'
ista: 'Hensel S. 2021. Curvature driven interface evolution: Uniqueness properties
of weak solution concepts. Institute of Science and Technology Austria.'
mla: 'Hensel, Sebastian. Curvature Driven Interface Evolution: Uniqueness Properties
of Weak Solution Concepts. Institute of Science and Technology Austria, 2021,
doi:10.15479/at:ista:10007.'
short: 'S. Hensel, Curvature Driven Interface Evolution: Uniqueness Properties of
Weak Solution Concepts, Institute of Science and Technology Austria, 2021.'
date_created: 2021-09-13T11:12:34Z
date_published: 2021-09-14T00:00:00Z
date_updated: 2023-09-07T13:30:45Z
day: '14'
ddc:
- '515'
degree_awarded: PhD
department:
- _id: GradSch
- _id: JuFi
doi: 10.15479/at:ista:10007
ec_funded: 1
file:
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checksum: c8475faaf0b680b4971f638f1db16347
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creator: shensel
date_created: 2021-09-13T11:03:24Z
date_updated: 2021-09-15T14:37:30Z
file_id: '10008'
file_name: thesis_final_Hensel.zip
file_size: 15022154
relation: source_file
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checksum: 1a609937aa5275452822f45f2da17f07
content_type: application/pdf
creator: shensel
date_created: 2021-09-13T14:18:56Z
date_updated: 2021-09-14T09:52:47Z
file_id: '10014'
file_name: thesis_final_Hensel.pdf
file_size: 6583638
relation: main_file
file_date_updated: 2021-09-15T14:37:30Z
has_accepted_license: '1'
language:
- iso: eng
month: '09'
oa: 1
oa_version: Published Version
page: '300'
project:
- _id: 2564DBCA-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '665385'
name: International IST Doctoral Program
- _id: 0aa76401-070f-11eb-9043-b5bb049fa26d
call_identifier: H2020
grant_number: '948819'
name: Bridging Scales in Random Materials
publication_identifier:
issn:
- 2663-337X
publication_status: published
publisher: Institute of Science and Technology Austria
related_material:
record:
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relation: part_of_dissertation
status: public
- id: '10013'
relation: part_of_dissertation
status: public
- id: '7489'
relation: part_of_dissertation
status: public
status: public
supervisor:
- first_name: Julian L
full_name: Fischer, Julian L
id: 2C12A0B0-F248-11E8-B48F-1D18A9856A87
last_name: Fischer
orcid: 0000-0002-0479-558X
title: 'Curvature driven interface evolution: Uniqueness properties of weak solution
concepts'
type: dissertation
user_id: c635000d-4b10-11ee-a964-aac5a93f6ac1
year: '2021'
...
---
_id: '10011'
abstract:
- lang: eng
text: We propose a new weak solution concept for (two-phase) mean curvature flow
which enjoys both (unconditional) existence and (weak-strong) uniqueness properties.
These solutions are evolving varifolds, just as in Brakke's formulation, but are
coupled to the phase volumes by a simple transport equation. First, we show that,
in the exact same setup as in Ilmanen's proof [J. Differential Geom. 38, 417-461,
(1993)], any limit point of solutions to the Allen-Cahn equation is a varifold
solution in our sense. Second, we prove that any calibrated flow in the sense
of Fischer et al. [arXiv:2003.05478] - and hence any classical solution to mean
curvature flow - is unique in the class of our new varifold solutions. This is
in sharp contrast to the case of Brakke flows, which a priori may disappear at
any given time and are therefore fatally non-unique. Finally, we propose an extension
of the solution concept to the multi-phase case which is at least guaranteed to
satisfy a weak-strong uniqueness principle.
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.
The content of this paper was developed and parts of it were written during a visit
of the first author to the Hausdorff Center of Mathematics (HCM), University of
Bonn. The hospitality and the support of HCM are gratefully acknowledged.
article_number: '2109.04233'
article_processing_charge: No
arxiv: 1
author:
- first_name: Sebastian
full_name: Hensel, Sebastian
id: 4D23B7DA-F248-11E8-B48F-1D18A9856A87
last_name: Hensel
orcid: 0000-0001-7252-8072
- first_name: Tim
full_name: Laux, Tim
last_name: Laux
citation:
ama: 'Hensel S, Laux T. A new varifold solution concept for mean curvature flow:
Convergence of the Allen-Cahn equation and weak-strong uniqueness. arXiv.
doi:10.48550/arXiv.2109.04233'
apa: 'Hensel, S., & Laux, T. (n.d.). A new varifold solution concept for mean
curvature flow: Convergence of the Allen-Cahn equation and weak-strong uniqueness.
arXiv. https://doi.org/10.48550/arXiv.2109.04233'
chicago: 'Hensel, Sebastian, and Tim Laux. “A New Varifold Solution Concept for
Mean Curvature Flow: Convergence of the Allen-Cahn Equation and Weak-Strong Uniqueness.”
ArXiv, n.d. https://doi.org/10.48550/arXiv.2109.04233.'
ieee: 'S. Hensel and T. Laux, “A new varifold solution concept for mean curvature
flow: Convergence of the Allen-Cahn equation and weak-strong uniqueness,” arXiv.
.'
ista: 'Hensel S, Laux T. A new varifold solution concept for mean curvature flow:
Convergence of the Allen-Cahn equation and weak-strong uniqueness. arXiv, 2109.04233.'
mla: 'Hensel, Sebastian, and Tim Laux. “A New Varifold Solution Concept for Mean
Curvature Flow: Convergence of the Allen-Cahn Equation and Weak-Strong Uniqueness.”
ArXiv, 2109.04233, doi:10.48550/arXiv.2109.04233.'
short: S. Hensel, T. Laux, ArXiv (n.d.).
date_created: 2021-09-13T12:17:10Z
date_published: 2021-09-09T00:00:00Z
date_updated: 2023-05-03T10:34:38Z
day: '09'
department:
- _id: JuFi
doi: 10.48550/arXiv.2109.04233
ec_funded: 1
external_id:
arxiv:
- '2109.04233'
keyword:
- Mean curvature flow
- gradient flows
- varifolds
- weak solutions
- weak-strong uniqueness
- calibrated geometry
- gradient-flow calibrations
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://arxiv.org/abs/2109.04233
month: '09'
oa: 1
oa_version: Preprint
project:
- _id: 0aa76401-070f-11eb-9043-b5bb049fa26d
call_identifier: H2020
grant_number: '948819'
name: Bridging Scales in Random Materials
publication: arXiv
publication_status: submitted
status: public
title: 'A new varifold solution concept for mean curvature flow: Convergence of the
Allen-Cahn equation and weak-strong uniqueness'
type: preprint
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
year: '2021'
...
---
_id: '10013'
abstract:
- lang: eng
text: We derive a weak-strong uniqueness principle for BV solutions to multiphase
mean curvature flow of triple line clusters in three dimensions. Our proof is
based on the explicit construction of a gradient-flow calibration in the sense
of the recent work of Fischer et al. [arXiv:2003.05478] for any such cluster.
This extends the two-dimensional construction to the three-dimensional case of
surfaces meeting along triple junctions.
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.
article_number: '2108.01733'
article_processing_charge: No
arxiv: 1
author:
- first_name: Sebastian
full_name: Hensel, Sebastian
id: 4D23B7DA-F248-11E8-B48F-1D18A9856A87
last_name: Hensel
orcid: 0000-0001-7252-8072
- first_name: Tim
full_name: Laux, Tim
last_name: Laux
citation:
ama: Hensel S, Laux T. Weak-strong uniqueness for the mean curvature flow of double
bubbles. arXiv. doi:10.48550/arXiv.2108.01733
apa: Hensel, S., & Laux, T. (n.d.). Weak-strong uniqueness for the mean curvature
flow of double bubbles. arXiv. https://doi.org/10.48550/arXiv.2108.01733
chicago: Hensel, Sebastian, and Tim Laux. “Weak-Strong Uniqueness for the Mean Curvature
Flow of Double Bubbles.” ArXiv, n.d. https://doi.org/10.48550/arXiv.2108.01733.
ieee: S. Hensel and T. Laux, “Weak-strong uniqueness for the mean curvature flow
of double bubbles,” arXiv. .
ista: Hensel S, Laux T. Weak-strong uniqueness for the mean curvature flow of double
bubbles. arXiv, 2108.01733.
mla: Hensel, Sebastian, and Tim Laux. “Weak-Strong Uniqueness for the Mean Curvature
Flow of Double Bubbles.” ArXiv, 2108.01733, doi:10.48550/arXiv.2108.01733.
short: S. Hensel, T. Laux, ArXiv (n.d.).
date_created: 2021-09-13T12:17:11Z
date_published: 2021-08-03T00:00:00Z
date_updated: 2023-09-07T13:30:45Z
day: '03'
department:
- _id: JuFi
doi: 10.48550/arXiv.2108.01733
ec_funded: 1
external_id:
arxiv:
- '2108.01733'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://arxiv.org/abs/2108.01733
month: '08'
oa: 1
oa_version: Preprint
project:
- _id: 0aa76401-070f-11eb-9043-b5bb049fa26d
call_identifier: H2020
grant_number: '948819'
name: Bridging Scales in Random Materials
publication: arXiv
publication_status: submitted
related_material:
record:
- id: '13043'
relation: later_version
status: public
- id: '10007'
relation: dissertation_contains
status: public
status: public
title: Weak-strong uniqueness for the mean curvature flow of double bubbles
type: preprint
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
year: '2021'
...
---
_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. Archive for Rational Mechanics and Analysis.
2020;236:967-1087. doi:10.1007/s00205-019-01486-2
apa: Fischer, J. L., & Hensel, S. (2020). Weak–strong uniqueness for the Navier–Stokes
equation for two fluids with surface tension. Archive for Rational Mechanics
and Analysis. Springer Nature. https://doi.org/10.1007/s00205-019-01486-2
chicago: Fischer, Julian L, and Sebastian Hensel. “Weak–Strong Uniqueness for the
Navier–Stokes Equation for Two Fluids with Surface Tension.” Archive for Rational
Mechanics and Analysis. Springer Nature, 2020. https://doi.org/10.1007/s00205-019-01486-2.
ieee: J. L. Fischer and S. Hensel, “Weak–strong uniqueness for the Navier–Stokes
equation for two fluids with surface tension,” Archive for Rational Mechanics
and Analysis, 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.” Archive for Rational Mechanics
and Analysis, vol. 236, Springer Nature, 2020, pp. 967–1087, doi:10.1007/s00205-019-01486-2.
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'
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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
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intvolume: ' 236'
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oa: 1
oa_version: Published Version
page: 967-1087
project:
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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:
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issn:
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publication_status: published
publisher: Springer Nature
quality_controlled: '1'
related_material:
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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
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: 236
year: '2020'
...
---
_id: '9196'
abstract:
- lang: eng
text: In order to provide a local description of a regular function in a small neighbourhood
of a point x, it is sufficient by Taylor’s theorem to know the value of the function
as well as all of its derivatives up to the required order at the point x itself.
In other words, one could say that a regular function is locally modelled by the
set of polynomials. The theory of regularity structures due to Hairer generalizes
this observation and provides an abstract setup, which in the application to singular
SPDE extends the set of polynomials by functionals constructed from, e.g., white
noise. In this context, the notion of Taylor polynomials is lifted to the notion
of so-called modelled distributions. The celebrated reconstruction theorem, which
in turn was inspired by Gubinelli’s \textit {sewing lemma}, is of paramount importance
for the theory. It enables one to reconstruct a modelled distribution as a true
distribution on Rd which is locally approximated by this extended set of models
or “monomials”. In the original work of Hairer, the error is measured by means
of Hölder norms. This was then generalized to the whole scale of Besov spaces
by Hairer and Labbé. It is the aim of this work to adapt the analytic part of
the theory of regularity structures to the scale of Triebel–Lizorkin spaces.
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Sebastian
full_name: Hensel, Sebastian
id: 4D23B7DA-F248-11E8-B48F-1D18A9856A87
last_name: Hensel
orcid: 0000-0001-7252-8072
- first_name: Tommaso
full_name: Rosati, Tommaso
last_name: Rosati
citation:
ama: Hensel S, Rosati T. Modelled distributions of Triebel–Lizorkin type. Studia
Mathematica. 2020;252(3):251-297. doi:10.4064/sm180411-11-2
apa: Hensel, S., & Rosati, T. (2020). Modelled distributions of Triebel–Lizorkin
type. Studia Mathematica. Instytut Matematyczny. https://doi.org/10.4064/sm180411-11-2
chicago: Hensel, Sebastian, and Tommaso Rosati. “Modelled Distributions of Triebel–Lizorkin
Type.” Studia Mathematica. Instytut Matematyczny, 2020. https://doi.org/10.4064/sm180411-11-2.
ieee: S. Hensel and T. Rosati, “Modelled distributions of Triebel–Lizorkin type,”
Studia Mathematica, vol. 252, no. 3. Instytut Matematyczny, pp. 251–297,
2020.
ista: Hensel S, Rosati T. 2020. Modelled distributions of Triebel–Lizorkin type.
Studia Mathematica. 252(3), 251–297.
mla: Hensel, Sebastian, and Tommaso Rosati. “Modelled Distributions of Triebel–Lizorkin
Type.” Studia Mathematica, vol. 252, no. 3, Instytut Matematyczny, 2020,
pp. 251–97, doi:10.4064/sm180411-11-2.
short: S. Hensel, T. Rosati, Studia Mathematica 252 (2020) 251–297.
date_created: 2021-02-25T08:55:03Z
date_published: 2020-03-01T00:00:00Z
date_updated: 2023-10-17T09:15:53Z
day: '01'
department:
- _id: JuFi
- _id: GradSch
doi: 10.4064/sm180411-11-2
external_id:
arxiv:
- '1709.05202'
isi:
- '000558100500002'
intvolume: ' 252'
isi: 1
issue: '3'
keyword:
- General Mathematics
language:
- iso: eng
month: '03'
oa_version: Preprint
page: 251-297
publication: Studia Mathematica
publication_identifier:
eissn:
- 1730-6337
issn:
- 0039-3223
publication_status: published
publisher: Instytut Matematyczny
quality_controlled: '1'
scopus_import: '1'
status: public
title: Modelled distributions of Triebel–Lizorkin type
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 252
year: '2020'
...
---
_id: '10012'
abstract:
- lang: eng
text: We prove that in the absence of topological changes, the notion of BV solutions
to planar multiphase mean curvature flow does not allow for a mechanism for (unphysical)
non-uniqueness. Our approach is based on the local structure of the energy landscape
near a classical evolution by mean curvature. Mean curvature flow being the gradient
flow of the surface energy functional, we develop a gradient-flow analogue of
the notion of calibrations. Just like the existence of a calibration guarantees
that one has reached a global minimum in the energy landscape, the existence of
a "gradient flow calibration" ensures that the route of steepest descent in the
energy landscape is unique and stable.
acknowledgement: Parts of the paper were written during the visit of the authors to
the Hausdorff Research Institute for Mathematics (HIM), University of Bonn, in the
framework of the trimester program “Evolution of Interfaces”. The support and the
hospitality of HIM are gratefully acknowledged. This project has received funding
from the European Union’s Horizon 2020 research and innovation programme under the
Marie Sklodowska-Curie Grant Agreement No. 665385.
article_number: '2003.05478'
article_processing_charge: No
arxiv: 1
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
- first_name: Tim
full_name: Laux, Tim
last_name: Laux
- first_name: Thilo
full_name: Simon, Thilo
last_name: Simon
citation:
ama: 'Fischer JL, Hensel S, Laux T, Simon T. The local structure of the energy landscape
in multiphase mean curvature flow: weak-strong uniqueness and stability of evolutions.
arXiv.'
apa: 'Fischer, J. L., Hensel, S., Laux, T., & Simon, T. (n.d.). The local structure
of the energy landscape in multiphase mean curvature flow: weak-strong uniqueness
and stability of evolutions. arXiv.'
chicago: 'Fischer, Julian L, Sebastian Hensel, Tim Laux, and Thilo Simon. “The Local
Structure of the Energy Landscape in Multiphase Mean Curvature Flow: Weak-Strong
Uniqueness and Stability of Evolutions.” ArXiv, n.d.'
ieee: 'J. L. Fischer, S. Hensel, T. Laux, and T. Simon, “The local structure of
the energy landscape in multiphase mean curvature flow: weak-strong uniqueness
and stability of evolutions,” arXiv. .'
ista: 'Fischer JL, Hensel S, Laux T, Simon T. The local structure of the energy
landscape in multiphase mean curvature flow: weak-strong uniqueness and stability
of evolutions. arXiv, 2003.05478.'
mla: 'Fischer, Julian L., et al. “The Local Structure of the Energy Landscape in
Multiphase Mean Curvature Flow: Weak-Strong Uniqueness and Stability of Evolutions.”
ArXiv, 2003.05478.'
short: J.L. Fischer, S. Hensel, T. Laux, T. Simon, ArXiv (n.d.).
date_created: 2021-09-13T12:17:11Z
date_published: 2020-03-11T00:00:00Z
date_updated: 2023-09-07T13:30:45Z
day: '11'
department:
- _id: JuFi
ec_funded: 1
external_id:
arxiv:
- '2003.05478'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://arxiv.org/abs/2003.05478
month: '03'
oa: 1
oa_version: Preprint
project:
- _id: 2564DBCA-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '665385'
name: International IST Doctoral Program
publication: arXiv
publication_status: submitted
related_material:
record:
- id: '10007'
relation: dissertation_contains
status: public
status: public
title: 'The local structure of the energy landscape in multiphase mean curvature flow:
weak-strong uniqueness and stability of evolutions'
type: preprint
user_id: 8b945eb4-e2f2-11eb-945a-df72226e66a9
year: '2020'
...