---
_id: '14587'
abstract:
- lang: eng
text: "This thesis concerns the application of variational methods to the study
of evolution problems arising in fluid mechanics and in material sciences. The
main focus is on weak-strong stability properties of some curvature driven interface
evolution problems, such as the two-phase Navier–Stokes flow with surface tension
and multiphase mean curvature flow, and on the phase-field approximation of the
latter. Furthermore, we discuss a variational approach to the study of a class
of doubly nonlinear wave equations.\r\nFirst, we consider the two-phase Navier–Stokes
flow with surface tension within a bounded domain. The two fluids are immiscible
and separated by a sharp interface, which intersects the boundary of the domain
at a constant contact angle of ninety degree. We devise a suitable concept of
varifolds solutions for the associated interface evolution problem and we establish
a weak-strong uniqueness principle in case of a two dimensional ambient space.
In order to focus on the boundary effects and on the singular geometry of the
evolving domains, we work for simplicity in the regime of same viscosities for
the two fluids.\r\nThe core of the thesis consists in the rigorous proof of the
convergence of the vectorial Allen-Cahn equation towards multiphase mean curvature
flow for a suitable class of multi- well potentials and for well-prepared initial
data. We even establish a rate of convergence. Our relative energy approach relies
on the concept of gradient-flow calibration for branching singularities in multiphase
mean curvature flow and thus enables us to overcome the limitations of other approaches.
To the best of the author’s knowledge, our result is the first quantitative and
unconditional one available in the literature for the vectorial/multiphase setting.\r\nThis
thesis also contains a first study of weak-strong stability for planar multiphase
mean curvature flow beyond the singularity resulting from a topology change. Previous
weak-strong results are indeed limited to time horizons before the first topology
change of the strong solution. We consider circular topology changes and we prove
weak-strong stability for BV solutions to planar multiphase mean curvature flow
beyond the associated singular times by dynamically adapting the strong solutions
to the weak one by means of a space-time shift.\r\nIn the context of interface
evolution problems, our proofs for the main results of this thesis are based on
the relative energy technique, relying on novel suitable notions of relative energy
functionals, which in particular measure the interface error. Our statements follow
from the resulting stability estimates for the relative energy associated to the
problem.\r\nAt last, we introduce a variational approach to the study of nonlinear
evolution problems. This approach hinges on the minimization of a parameter dependent
family of convex functionals over entire trajectories, known as Weighted Inertia-Dissipation-Energy
(WIDE) functionals. We consider a class of doubly nonlinear wave equations and
establish the convergence, up to subsequences, of the associated WIDE minimizers
to a solution of the target problem as the parameter goes to zero."
acknowledgement: The research projects contained in this thesis have received funding
from the European Research Council (ERC) under the European Union’s Horizon 2020
research and innovation programme (grant agreement No 948819).
alternative_title:
- ISTA Thesis
article_processing_charge: No
author:
- first_name: Alice
full_name: Marveggio, Alice
id: 25647992-AA84-11E9-9D75-8427E6697425
last_name: Marveggio
citation:
ama: Marveggio A. Weak-strong stability and phase-field approximation of interface
evolution problems in fluid mechanics and in material sciences. 2023. doi:10.15479/at:ista:14587
apa: Marveggio, A. (2023). Weak-strong stability and phase-field approximation
of interface evolution problems in fluid mechanics and in material sciences.
Institute of Science and Technology Austria. https://doi.org/10.15479/at:ista:14587
chicago: Marveggio, Alice. “Weak-Strong Stability and Phase-Field Approximation
of Interface Evolution Problems in Fluid Mechanics and in Material Sciences.”
Institute of Science and Technology Austria, 2023. https://doi.org/10.15479/at:ista:14587.
ieee: A. Marveggio, “Weak-strong stability and phase-field approximation of interface
evolution problems in fluid mechanics and in material sciences,” Institute of
Science and Technology Austria, 2023.
ista: Marveggio A. 2023. Weak-strong stability and phase-field approximation of
interface evolution problems in fluid mechanics and in material sciences. Institute
of Science and Technology Austria.
mla: Marveggio, Alice. Weak-Strong Stability and Phase-Field Approximation of
Interface Evolution Problems in Fluid Mechanics and in Material Sciences.
Institute of Science and Technology Austria, 2023, doi:10.15479/at:ista:14587.
short: A. Marveggio, Weak-Strong Stability and Phase-Field Approximation of Interface
Evolution Problems in Fluid Mechanics and in Material Sciences, Institute of Science
and Technology Austria, 2023.
date_created: 2023-11-21T11:41:05Z
date_published: 2023-11-21T00:00:00Z
date_updated: 2023-11-30T13:25:03Z
day: '21'
ddc:
- '515'
degree_awarded: PhD
department:
- _id: GradSch
- _id: JuFi
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ec_funded: 1
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- _id: 0aa76401-070f-11eb-9043-b5bb049fa26d
call_identifier: H2020
grant_number: '948819'
name: Bridging Scales in Random Materials
publication_identifier:
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- 2663 - 337X
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publisher: Institute of Science and Technology Austria
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full_name: Fischer, Julian L
id: 2C12A0B0-F248-11E8-B48F-1D18A9856A87
last_name: Fischer
orcid: 0000-0002-0479-558X
title: Weak-strong stability and phase-field approximation of interface evolution
problems in fluid mechanics and in material sciences
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...
---
_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
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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
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relation: part_of_dissertation
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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
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