---
_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.'
corr_author: '1'
date_created: 2021-09-13T11:12:34Z
date_published: 2021-09-14T00:00:00Z
date_updated: 2025-07-10T11:54:41Z
day: '14'
ddc:
- '515'
degree_awarded: PhD
department:
- _id: GradSch
- _id: JuFi
doi: 10.15479/at:ista:10007
ec_funded: 1
file:
- access_level: closed
checksum: c8475faaf0b680b4971f638f1db16347
content_type: application/x-zip-compressed
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
- access_level: open_access
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:
- id: '10012'
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: 2DF688A6-F248-11E8-B48F-1D18A9856A87
year: '2021'
...