---
_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
doi: 10.15479/at:ista:14587
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:
issn:
- 2663 - 337X
publication_status: published
publisher: Institute of Science and Technology Austria
related_material:
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- id: '11842'
relation: part_of_dissertation
status: public
- id: '14597'
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: Weak-strong stability and phase-field approximation of interface evolution
problems in fluid mechanics and in material sciences
tmp:
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type: dissertation
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...
---
_id: '14597'
abstract:
- lang: eng
text: "Phase-field models such as the Allen-Cahn equation may give rise to the formation
and evolution of geometric shapes, a phenomenon that may be analyzed rigorously
in suitable scaling regimes. In its sharp-interface limit, the vectorial Allen-Cahn
equation with a potential with N≥3 distinct minima has been conjectured to describe
the evolution of branched interfaces by multiphase mean curvature flow.\r\nIn
the present work, we give a rigorous proof for this statement in two and three
ambient dimensions and for a suitable class of potentials: As long as a strong
solution to multiphase mean curvature flow exists, solutions to the vectorial
Allen-Cahn equation with well-prepared initial data converge towards multiphase
mean curvature flow in the limit of vanishing interface width parameter ε↘0. We
even establish the rate of convergence O(ε1/2).\r\nOur approach is based on the
gradient flow structure of the Allen-Cahn equation and its limiting motion: Building
on the recent concept of \"gradient flow calibrations\" for multiphase mean curvature
flow, we introduce a notion of relative entropy for the vectorial Allen-Cahn equation
with multi-well potential. This enables us to overcome the limitations of other
approaches, e.g. avoiding the need for a stability analysis of the Allen-Cahn
operator or additional convergence hypotheses for the energy at positive times."
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: Alice
full_name: Marveggio, Alice
id: 25647992-AA84-11E9-9D75-8427E6697425
last_name: Marveggio
citation:
ama: Fischer JL, Marveggio A. Quantitative convergence of the vectorial Allen-Cahn
equation towards multiphase mean curvature flow. arXiv. doi:10.48550/ARXIV.2203.17143
apa: Fischer, J. L., & Marveggio, A. (n.d.). Quantitative convergence of the
vectorial Allen-Cahn equation towards multiphase mean curvature flow. arXiv.
https://doi.org/10.48550/ARXIV.2203.17143
chicago: Fischer, Julian L, and Alice Marveggio. “Quantitative Convergence of the
Vectorial Allen-Cahn Equation towards Multiphase Mean Curvature Flow.” ArXiv,
n.d. https://doi.org/10.48550/ARXIV.2203.17143.
ieee: J. L. Fischer and A. Marveggio, “Quantitative convergence of the vectorial
Allen-Cahn equation towards multiphase mean curvature flow,” arXiv. .
ista: Fischer JL, Marveggio A. Quantitative convergence of the vectorial Allen-Cahn
equation towards multiphase mean curvature flow. arXiv, 10.48550/ARXIV.2203.17143.
mla: Fischer, Julian L., and Alice Marveggio. “Quantitative Convergence of the Vectorial
Allen-Cahn Equation towards Multiphase Mean Curvature Flow.” ArXiv, doi:10.48550/ARXIV.2203.17143.
short: J.L. Fischer, A. Marveggio, ArXiv (n.d.).
date_created: 2023-11-23T09:30:02Z
date_published: 2022-03-31T00:00:00Z
date_updated: 2023-11-30T13:25:02Z
day: '31'
department:
- _id: JuFi
doi: 10.48550/ARXIV.2203.17143
ec_funded: 1
external_id:
arxiv:
- '2203.17143'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://arxiv.org/abs/2203.17143
month: '03'
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:
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- id: '14587'
relation: dissertation_contains
status: public
status: public
title: Quantitative convergence of the vectorial Allen-Cahn equation towards multiphase
mean curvature flow
type: preprint
user_id: 8b945eb4-e2f2-11eb-945a-df72226e66a9
year: '2022'
...
---
_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'
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checksum: 75c5f286300e6f0539cf57b4dba108d5
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creator: cchlebak
date_created: 2022-08-16T06:55:22Z
date_updated: 2022-08-16T06:55:22Z
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file_name: 2022_JMathFluidMech_Hensel.pdf
file_size: 2045570
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file_date_updated: 2022-08-16T06:55:22Z
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isi: 1
issue: '3'
language:
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license: https://creativecommons.org/licenses/by/4.0/
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: '8792'
abstract:
- lang: eng
text: This paper is concerned with a non-isothermal Cahn-Hilliard model based on
a microforce balance. The model was derived by A. Miranville and G. Schimperna
starting from the two fundamental laws of Thermodynamics, following M. Gurtin's
two-scale approach. The main working assumptions are made on the behaviour of
the heat flux as the absolute temperature tends to zero and to infinity. A suitable
Ginzburg-Landau free energy is considered. Global-in-time existence for the initial-boundary
value problem associated to the entropy formulation and, in a subcase, also to
the weak formulation of the model is proved by deriving suitable a priori estimates
and by showing weak sequential stability of families of approximating solutions.
At last, some highlights are given regarding a possible approximation scheme compatible
with the a-priori estimates available for the system.
acknowledgement: G. Schimperna has been partially supported by GNAMPA (Gruppo Nazionale
per l'Analisi Matematica, la Probabilità e le loro Applicazioni) of INdAM (Istituto
Nazionale di Alta Matematica).
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Alice
full_name: Marveggio, Alice
id: 25647992-AA84-11E9-9D75-8427E6697425
last_name: Marveggio
- first_name: Giulio
full_name: Schimperna, Giulio
last_name: Schimperna
citation:
ama: Marveggio A, Schimperna G. On a non-isothermal Cahn-Hilliard model based on
a microforce balance. Journal of Differential Equations. 2021;274(2):924-970.
doi:10.1016/j.jde.2020.10.030
apa: Marveggio, A., & Schimperna, G. (2021). On a non-isothermal Cahn-Hilliard
model based on a microforce balance. Journal of Differential Equations.
Elsevier. https://doi.org/10.1016/j.jde.2020.10.030
chicago: Marveggio, Alice, and Giulio Schimperna. “On a Non-Isothermal Cahn-Hilliard
Model Based on a Microforce Balance.” Journal of Differential Equations.
Elsevier, 2021. https://doi.org/10.1016/j.jde.2020.10.030.
ieee: A. Marveggio and G. Schimperna, “On a non-isothermal Cahn-Hilliard model based
on a microforce balance,” Journal of Differential Equations, vol. 274,
no. 2. Elsevier, pp. 924–970, 2021.
ista: Marveggio A, Schimperna G. 2021. On a non-isothermal Cahn-Hilliard model based
on a microforce balance. Journal of Differential Equations. 274(2), 924–970.
mla: Marveggio, Alice, and Giulio Schimperna. “On a Non-Isothermal Cahn-Hilliard
Model Based on a Microforce Balance.” Journal of Differential Equations,
vol. 274, no. 2, Elsevier, 2021, pp. 924–70, doi:10.1016/j.jde.2020.10.030.
short: A. Marveggio, G. Schimperna, Journal of Differential Equations 274 (2021)
924–970.
date_created: 2020-11-22T23:01:26Z
date_published: 2021-02-15T00:00:00Z
date_updated: 2023-08-04T11:12:16Z
day: '15'
department:
- _id: JuFi
doi: 10.1016/j.jde.2020.10.030
external_id:
arxiv:
- '2004.02618'
isi:
- '000600845300023'
intvolume: ' 274'
isi: 1
issue: '2'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://arxiv.org/abs/2004.02618
month: '02'
oa: 1
oa_version: Preprint
page: 924-970
publication: Journal of Differential Equations
publication_identifier:
eissn:
- '10902732'
issn:
- '00220396'
publication_status: published
publisher: Elsevier
quality_controlled: '1'
scopus_import: '1'
status: public
title: On a non-isothermal Cahn-Hilliard model based on a microforce balance
type: journal_article
user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
volume: 274
year: '2021'
...