---
OA_place: repository
OA_type: green
_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_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. A new varifold solution concept for mean curvature flow:
Convergence of the Allen-Cahn equation and weak-strong uniqueness. Journal
of Differential Geometry. 2025;130:209-268. doi:10.4310/jdg/1747065796'
apa: 'Hensel, S., & Laux, T. (2025). A new varifold solution concept for mean
curvature flow: Convergence of the Allen-Cahn equation and weak-strong uniqueness.
Journal of Differential Geometry. International Press of Boston. https://doi.org/10.4310/jdg/1747065796'
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.”
Journal of Differential Geometry. International Press of Boston, 2025.
https://doi.org/10.4310/jdg/1747065796.'
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,” Journal
of Differential Geometry, vol. 130. International Press of Boston, pp. 209–268,
2025.'
ista: 'Hensel S, Laux T. 2025. A new varifold solution concept for mean curvature
flow: Convergence of the Allen-Cahn equation and weak-strong uniqueness. Journal
of Differential Geometry. 130, 209–268.'
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.”
Journal of Differential Geometry, vol. 130, International Press of Boston,
2025, pp. 209–68, doi:10.4310/jdg/1747065796.'
short: S. Hensel, T. Laux, Journal of Differential Geometry 130 (2025) 209–268.
corr_author: '1'
das_tickbox: '1'
date_created: 2021-09-13T12:17:10Z
date_published: 2025-05-01T00:00:00Z
date_updated: 2026-07-06T13:37:21Z
day: '01'
department:
- _id: JuFi
doi: 10.4310/jdg/1747065796
ec_funded: 1
external_id:
arxiv:
- '2109.04233'
fulldoi: https://doi.org/10.4310/jdg/1747065796
intvolume: ' 130'
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: '05'
oa: 1
oa_version: Preprint
page: 209-268
project:
- _id: 0aa76401-070f-11eb-9043-b5bb049fa26d
call_identifier: H2020
grant_number: '948819'
name: Bridging Scales in Random Materials
publication: Journal of Differential Geometry
publication_identifier:
eissn:
- 1945-743X
issn:
- 0022-040X
publication_status: published
publisher: International Press of Boston
quality_controlled: '1'
scopus_import: '1'
status: public
title: 'A new varifold solution concept for mean curvature flow: Convergence of the
Allen-Cahn equation and weak-strong uniqueness'
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 130
year: '2025'
...
---
OA_place: repository
OA_type: green
_id: '22026'
abstract:
- lang: eng
text: We address two pressing questions in the theory of the Korteweg–de Vries (KdV)
equation. First, we show the uniqueness of solutions to KdV that are merely bounded,
without any further decay, regularity, periodicity, or almost periodicity assumptions.
The second question, emphasized by Deift, regards whether almost periodic initial
data leads to almost periodic solutions to KdV. Building on the new observation
that this is false for the Airy equation, we construct an example of almost periodic
initial data whose KdV evolution remains bounded, but fails to be almost periodic
at a later time. Our uniqueness result ensures that the solution constructed is
the unique development of this initial data.
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Andreia
full_name: Chapouto, Andreia
last_name: Chapouto
- first_name: Rowan
full_name: Killip, Rowan
last_name: Killip
- first_name: Monica
full_name: Visan, Monica
id: 056daca0-b8d1-11f0-964f-f91054abf8ca
last_name: Visan
citation:
ama: 'Chapouto A, Killip R, Vişan M. Bounded solutions of KdV: Uniqueness and the
loss of almost periodicity. Duke Mathematical Journal. 2024;173(7):1227-1267.
doi:10.1215/00127094-2023-0035'
apa: 'Chapouto, A., Killip, R., & Vişan, M. (2024). Bounded solutions of KdV:
Uniqueness and the loss of almost periodicity. Duke Mathematical Journal.
Duke University Press. https://doi.org/10.1215/00127094-2023-0035'
chicago: 'Chapouto, Andreia, Rowan Killip, and Monica Vişan. “Bounded Solutions
of KdV: Uniqueness and the Loss of Almost Periodicity.” Duke Mathematical Journal.
Duke University Press, 2024. https://doi.org/10.1215/00127094-2023-0035.'
ieee: 'A. Chapouto, R. Killip, and M. Vişan, “Bounded solutions of KdV: Uniqueness
and the loss of almost periodicity,” Duke Mathematical Journal, vol. 173,
no. 7. Duke University Press, pp. 1227–1267, 2024.'
ista: 'Chapouto A, Killip R, Vişan M. 2024. Bounded solutions of KdV: Uniqueness
and the loss of almost periodicity. Duke Mathematical Journal. 173(7), 1227–1267.'
mla: 'Chapouto, Andreia, et al. “Bounded Solutions of KdV: Uniqueness and the Loss
of Almost Periodicity.” Duke Mathematical Journal, vol. 173, no. 7, Duke
University Press, 2024, pp. 1227–67, doi:10.1215/00127094-2023-0035.'
short: A. Chapouto, R. Killip, M. Vişan, Duke Mathematical Journal 173 (2024) 1227–1267.
das_tickbox: '1'
date_created: 2026-06-19T07:34:05Z
date_published: 2024-05-15T00:00:00Z
date_updated: 2026-06-22T10:32:25Z
day: '15'
doi: 10.1215/00127094-2023-0035
extern: '1'
external_id:
arxiv:
- '2209.07501'
fulldoi: https://doi.org/10.1215/00127094-2023-0035
intvolume: ' 173'
issue: '7'
keyword:
- Almost-periodic solutions
- Korteweg–de Vries
- unconditional uniqueness
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://doi.org/10.48550/arXiv.2209.07501
month: '05'
oa: 1
oa_version: Preprint
page: 1227-1267
publication: Duke Mathematical Journal
publication_identifier:
issn:
- 0012-7094
publication_status: published
publisher: Duke University Press
quality_controlled: '1'
scopus_import: '1'
status: public
title: 'Bounded solutions of KdV: Uniqueness and the loss of almost periodicity'
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 173
year: '2024'
...
---
_id: '10005'
abstract:
- lang: eng
text: We study systems of nonlinear partial differential equations of parabolic
type, in which the elliptic operator is replaced by the first-order divergence
operator acting on a flux function, which is related to the spatial gradient of
the unknown through an additional implicit equation. This setting, broad enough
in terms of applications, significantly expands the paradigm of nonlinear parabolic
problems. Formulating four conditions concerning the form of the implicit equation,
we first show that these conditions describe a maximal monotone p-coercive graph.
We then establish the global-in-time and large-data existence of a (weak) solution
and its uniqueness. To this end, we adopt and significantly generalize Minty’s
method of monotone mappings. A unified theory, containing several novel tools,
is developed in a way to be tractable from the point of view of numerical approximations.
acknowledgement: "M. Bulíček and J. Málek acknowledge the support of the project No.
18-12719S financed by the Czech\r\nScience foundation (GAČR). E. Maringová acknowledges
support from Charles University Research program \r\nUNCE/SCI/023, the grant SVV-2020-260583
by the Ministry of Education, Youth and Sports, Czech Republic\r\nand from the Austrian
Science Fund (FWF), grants P30000, W1245, and F65. M. Bulíček and J. Málek are\r\nmembers
of the Nečas Center for Mathematical Modelling.\r\n"
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Miroslav
full_name: Bulíček, Miroslav
last_name: Bulíček
- first_name: Erika
full_name: Maringová, Erika
id: dbabca31-66eb-11eb-963a-fb9c22c880b4
last_name: Maringová
- first_name: Josef
full_name: Málek, Josef
last_name: Málek
citation:
ama: Bulíček M, Maringová E, Málek J. On nonlinear problems of parabolic type with
implicit constitutive equations involving flux. Mathematical Models and Methods
in Applied Sciences. 2021;31(09). doi:10.1142/S0218202521500457
apa: Bulíček, M., Maringová, E., & Málek, J. (2021). On nonlinear problems of
parabolic type with implicit constitutive equations involving flux. Mathematical
Models and Methods in Applied Sciences. World Scientific Publishing. https://doi.org/10.1142/S0218202521500457
chicago: Bulíček, Miroslav, Erika Maringová, and Josef Málek. “On Nonlinear Problems
of Parabolic Type with Implicit Constitutive Equations Involving Flux.” Mathematical
Models and Methods in Applied Sciences. World Scientific Publishing, 2021.
https://doi.org/10.1142/S0218202521500457.
ieee: M. Bulíček, E. Maringová, and J. Málek, “On nonlinear problems of parabolic
type with implicit constitutive equations involving flux,” Mathematical Models
and Methods in Applied Sciences, vol. 31, no. 09. World Scientific Publishing,
2021.
ista: Bulíček M, Maringová E, Málek J. 2021. On nonlinear problems of parabolic
type with implicit constitutive equations involving flux. Mathematical Models
and Methods in Applied Sciences. 31(09).
mla: Bulíček, Miroslav, et al. “On Nonlinear Problems of Parabolic Type with Implicit
Constitutive Equations Involving Flux.” Mathematical Models and Methods in
Applied Sciences, vol. 31, no. 09, World Scientific Publishing, 2021, doi:10.1142/S0218202521500457.
short: M. Bulíček, E. Maringová, J. Málek, Mathematical Models and Methods in Applied
Sciences 31 (2021).
date_created: 2021-09-12T22:01:25Z
date_published: 2021-08-25T00:00:00Z
date_updated: 2026-08-18T07:47:39Z
day: '25'
department:
- _id: JuFi
doi: 10.1142/S0218202521500457
external_id:
arxiv:
- '2009.06917'
isi:
- '000722222900004'
fulldoi: https://doi.org/10.1142/S0218202521500457
intvolume: ' 31'
isi: 1
issue: '09'
keyword:
- Nonlinear parabolic systems
- implicit constitutive theory
- weak solutions
- existence
- uniqueness
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://arxiv.org/abs/2009.06917
month: '08'
oa: 1
oa_version: Preprint
project:
- _id: fc31cba2-9c52-11eb-aca3-ff467d239cd2
grant_number: F6504
name: Taming Complexity in Partial Differential Systems
publication: Mathematical Models and Methods in Applied Sciences
publication_identifier:
eissn:
- 1793-6314
issn:
- 0218-2025
publication_status: published
publisher: World Scientific Publishing
quality_controlled: '1'
scopus_import: '1'
status: public
title: On nonlinear problems of parabolic type with implicit constitutive equations
involving flux
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 31
year: '2021'
...
---
OA_type: closed access
_id: '22510'
abstract:
- lang: eng
text: This study advances mechanistic interpretation of predictability challenges
in hydro-geomorphology related to the role of soil moisture spatial variability.
Using model formulations describing the physics of overland flow, variably saturated
subsurface flow, and erosion and sediment transport, this study explores (1) why
a basin with the same mean soil moisture can exhibit distinctly different spatial
moisture distributions, (2) whether these varying distributions lead to non-unique
hydro-geomorphic responses, and (3) what controls non-uniqueness in relation to
the response type. Two sets of numerical experiments are carried out with two
physically-based models, HYDRUS and tRIBS+VEGGIE+FEaST, and their outputs are
analyzed with respect to pre-storm moisture state. The results demonstrate that
distinct spatial moisture distributions for the same mean wetness arise because
near-surface soil moisture dynamics exhibit different degrees of coupling with
deeper-soil moisture and the process of subsurface drainage. The consequences
of such variations are different depending on the type of hydrological response.
Specifically, if the predominant runoff response is of infiltration excess type,
the degree of non-uniqueness is related to the spatial distribution of near-surface
moisture. If runoff is governed by subsurface stormflow, the extent of deep moisture
contributing area and its “readiness to drain” determine the response characteristics.
Because the processes of erosion and sediment transport superimpose additional
controls over factors governing runoff generation and overland flow, non-uniqueness
of the geomorphic response can be highly dampened or enhanced. The explanation
is sediment composed by multi-size particles can alternate states of mobilization
or surface shielding and the transient behavior is inherently intertwined with
the availability of mobile particles. We conclude that complex nonlinear dynamics
of hydro-geomorphic processes are inherent expressions of physical interactions.
As complete knowledge of watershed properties, states, or forcings will always
present the ultimate, if ever resolvable, challenge, deterministic predictability
will remain handicapped. Coupling of uncertainty quantification methods and space-time
physics-based approaches will need to evolve to facilitate mechanistic interpretations
and informed practical applications.
article_processing_charge: No
article_type: original
author:
- first_name: Jongho
full_name: Kim, Jongho
last_name: Kim
- first_name: M. Chase
full_name: Dwelle, M. Chase
last_name: Dwelle
- first_name: Stephanie K.
full_name: Kampf, Stephanie K.
last_name: Kampf
- first_name: Simone
full_name: Fatichi, Simone
id: cf8e546b-a9b0-11f0-a43b-aa89ed1b56d6
last_name: Fatichi
- first_name: Valeriy Y.
full_name: Ivanov, Valeriy Y.
last_name: Ivanov
citation:
ama: Kim J, Dwelle MC, Kampf SK, Fatichi S, Ivanov VY. On the non-uniqueness of
the hydro-geomorphic responses in a zero-order catchment with respect to soil
moisture. Advances in Water Resources. 2016;92:73-89. doi:10.1016/j.advwatres.2016.03.019
apa: Kim, J., Dwelle, M. C., Kampf, S. K., Fatichi, S., & Ivanov, V. Y. (2016).
On the non-uniqueness of the hydro-geomorphic responses in a zero-order catchment
with respect to soil moisture. Advances in Water Resources. Elsevier. https://doi.org/10.1016/j.advwatres.2016.03.019
chicago: Kim, Jongho, M. Chase Dwelle, Stephanie K. Kampf, Simone Fatichi, and Valeriy
Y. Ivanov. “On the Non-Uniqueness of the Hydro-Geomorphic Responses in a Zero-Order
Catchment with Respect to Soil Moisture.” Advances in Water Resources.
Elsevier, 2016. https://doi.org/10.1016/j.advwatres.2016.03.019.
ieee: J. Kim, M. C. Dwelle, S. K. Kampf, S. Fatichi, and V. Y. Ivanov, “On the non-uniqueness
of the hydro-geomorphic responses in a zero-order catchment with respect to soil
moisture,” Advances in Water Resources, vol. 92. Elsevier, pp. 73–89, 2016.
ista: Kim J, Dwelle MC, Kampf SK, Fatichi S, Ivanov VY. 2016. On the non-uniqueness
of the hydro-geomorphic responses in a zero-order catchment with respect to soil
moisture. Advances in Water Resources. 92, 73–89.
mla: Kim, Jongho, et al. “On the Non-Uniqueness of the Hydro-Geomorphic Responses
in a Zero-Order Catchment with Respect to Soil Moisture.” Advances in Water
Resources, vol. 92, Elsevier, 2016, pp. 73–89, doi:10.1016/j.advwatres.2016.03.019.
short: J. Kim, M.C. Dwelle, S.K. Kampf, S. Fatichi, V.Y. Ivanov, Advances in Water
Resources 92 (2016) 73–89.
das_tickbox: '1'
date_created: 2026-07-27T12:30:24Z
date_published: 2016-07-01T00:00:00Z
date_updated: 2026-08-06T07:50:36Z
day: '01'
doi: 10.1016/j.advwatres.2016.03.019
extern: '1'
fulldoi: https://doi.org/10.1016/j.advwatres.2016.03.019
intvolume: ' 92'
keyword:
- Soil moisture
- Spatial heterogeneity
- Hydrological response
- Geomorphic response
- Non-uniqueness
- Hydraulic connectivity
language:
- iso: eng
month: '07'
oa_version: None
page: 73-89
publication: Advances in Water Resources
publication_identifier:
issn:
- 0309-1708
publication_status: published
publisher: Elsevier
quality_controlled: '1'
scopus_import: '1'
status: public
title: On the non-uniqueness of the hydro-geomorphic responses in a zero-order catchment
with respect to soil moisture
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 92
year: '2016'
...