---
_id: '14042'
abstract:
- lang: eng
text: Long-time and large-data existence of weak solutions for initial- and boundary-value
problems concerning three-dimensional flows of incompressible fluids is nowadays
available not only for Navier–Stokes fluids but also for various fluid models
where the relation between the Cauchy stress tensor and the symmetric part of
the velocity gradient is nonlinear. The majority of such studies however concerns
models where such a dependence is explicit (the stress is a function of the velocity
gradient), which makes the class of studied models unduly restrictive. The same
concerns boundary conditions, or more precisely the slipping mechanisms on the
boundary, where the no-slip is still the most preferred condition considered in
the literature. Our main objective is to develop a robust mathematical theory
for unsteady internal flows of implicitly constituted incompressible fluids with
implicit relations between the tangential projections of the velocity and the
normal traction on the boundary. The theory covers numerous rheological models
used in chemistry, biorheology, polymer and food industry as well as in geomechanics.
It also includes, as special cases, nonlinear slip as well as stick–slip boundary
conditions. Unlike earlier studies, the conditions characterizing admissible classes
of constitutive equations are expressed by means of tools of elementary calculus.
In addition, a fully constructive proof (approximation scheme) is incorporated.
Finally, we focus on the question of uniqueness of such weak solutions.
acknowledgement: "M. Bulíček and J. Málek acknowledge the support of the project No.
20-11027X financed by the Czech Science foundation (GAČR). M. Bulíček and J. Málek
are members of the Nečas Center for Mathematical Modelling.\r\nOpen access publishing
supported by the National Technical Library in Prague."
article_number: '72'
article_processing_charge: Yes (via OA deal)
article_type: original
arxiv: 1
author:
- first_name: Miroslav
full_name: Bulíček, Miroslav
last_name: Bulíček
- first_name: Josef
full_name: Málek, Josef
last_name: Málek
- first_name: Erika
full_name: Maringová, Erika
id: dbabca31-66eb-11eb-963a-fb9c22c880b4
last_name: Maringová
citation:
ama: Bulíček M, Málek J, Maringová E. On unsteady internal flows of incompressible
fluids characterized by implicit constitutive equations in the bulk and on the
boundary. Journal of Mathematical Fluid Mechanics. 2023;25(3). doi:10.1007/s00021-023-00803-w
apa: Bulíček, M., Málek, J., & Maringová, E. (2023). On unsteady internal flows
of incompressible fluids characterized by implicit constitutive equations in the
bulk and on the boundary. Journal of Mathematical Fluid Mechanics. Springer
Nature. https://doi.org/10.1007/s00021-023-00803-w
chicago: Bulíček, Miroslav, Josef Málek, and Erika Maringová. “On Unsteady Internal
Flows of Incompressible Fluids Characterized by Implicit Constitutive Equations
in the Bulk and on the Boundary.” Journal of Mathematical Fluid Mechanics.
Springer Nature, 2023. https://doi.org/10.1007/s00021-023-00803-w.
ieee: M. Bulíček, J. Málek, and E. Maringová, “On unsteady internal flows of incompressible
fluids characterized by implicit constitutive equations in the bulk and on the
boundary,” Journal of Mathematical Fluid Mechanics, vol. 25, no. 3. Springer
Nature, 2023.
ista: Bulíček M, Málek J, Maringová E. 2023. On unsteady internal flows of incompressible
fluids characterized by implicit constitutive equations in the bulk and on the
boundary. Journal of Mathematical Fluid Mechanics. 25(3), 72.
mla: Bulíček, Miroslav, et al. “On Unsteady Internal Flows of Incompressible Fluids
Characterized by Implicit Constitutive Equations in the Bulk and on the Boundary.”
Journal of Mathematical Fluid Mechanics, vol. 25, no. 3, 72, Springer Nature,
2023, doi:10.1007/s00021-023-00803-w.
short: M. Bulíček, J. Málek, E. Maringová, Journal of Mathematical Fluid Mechanics
25 (2023).
date_created: 2023-08-13T22:01:13Z
date_published: 2023-08-01T00:00:00Z
date_updated: 2023-12-13T12:08:08Z
day: '01'
ddc:
- '510'
department:
- _id: JuFi
doi: 10.1007/s00021-023-00803-w
external_id:
arxiv:
- '2301.12834'
isi:
- '001040354900001'
file:
- access_level: open_access
checksum: c549cd8f0dd02ed60477a05ca045f481
content_type: application/pdf
creator: dernst
date_created: 2023-08-14T07:24:17Z
date_updated: 2023-08-14T07:24:17Z
file_id: '14046'
file_name: 2023_JourMathFluidMech_Bulicek.pdf
file_size: 845748
relation: main_file
success: 1
file_date_updated: 2023-08-14T07:24:17Z
has_accepted_license: '1'
intvolume: ' 25'
isi: 1
issue: '3'
language:
- iso: eng
month: '08'
oa: 1
oa_version: Published Version
publication: Journal of Mathematical Fluid Mechanics
publication_identifier:
eissn:
- 1422-6952
issn:
- 1422-6928
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: On unsteady internal flows of incompressible fluids characterized by implicit
constitutive equations in the bulk and on the boundary
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: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 25
year: '2023'
...
---
_id: '10575'
abstract:
- lang: eng
text: The choice of the boundary conditions in mechanical problems has to reflect
the interaction of the considered material with the surface. Still the assumption
of the no-slip condition is preferred in order to avoid boundary terms in the
analysis and slipping effects are usually overlooked. Besides the “static slip
models”, there are phenomena that are not accurately described by them, e.g. at
the moment when the slip changes rapidly, the wall shear stress and the slip can
exhibit a sudden overshoot and subsequent relaxation. When these effects become
significant, the so-called dynamic slip phenomenon occurs. We develop a mathematical
analysis of Navier–Stokes-like problems with a dynamic slip boundary condition,
which requires a proper generalization of the Gelfand triplet and the corresponding
function space setting.
acknowledgement: The research of A. Abbatiello is supported by Einstein Foundation,
Berlin. A. Abbatiello is also member of the Italian National Group for the Mathematical
Physics (GNFM) of INdAM. M. Bulíček acknowledges the support of the project No.
20-11027X financed by Czech Science Foundation (GACR). M. Bulíček is member of the
Jindřich Nečas Center for Mathematical Modelling. E. Maringová acknowledges support
from Charles University Research program UNCE/SCI/023, the grant SVV-2020-260583
by the Ministry of Education, Youth and Sports, Czech Republic and from the Austrian
Science Fund (FWF), grants P30000, W1245, and F65.
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Anna
full_name: Abbatiello, Anna
last_name: Abbatiello
- 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
citation:
ama: Abbatiello A, Bulíček M, Maringová E. On the dynamic slip boundary condition
for Navier-Stokes-like problems. Mathematical Models and Methods in Applied
Sciences. 2021;31(11):2165-2212. doi:10.1142/S0218202521500470
apa: Abbatiello, A., Bulíček, M., & Maringová, E. (2021). On the dynamic slip
boundary condition for Navier-Stokes-like problems. Mathematical Models and
Methods in Applied Sciences. World Scientific Publishing. https://doi.org/10.1142/S0218202521500470
chicago: Abbatiello, Anna, Miroslav Bulíček, and Erika Maringová. “On the Dynamic
Slip Boundary Condition for Navier-Stokes-like Problems.” Mathematical Models
and Methods in Applied Sciences. World Scientific Publishing, 2021. https://doi.org/10.1142/S0218202521500470.
ieee: A. Abbatiello, M. Bulíček, and E. Maringová, “On the dynamic slip boundary
condition for Navier-Stokes-like problems,” Mathematical Models and Methods
in Applied Sciences, vol. 31, no. 11. World Scientific Publishing, pp. 2165–2212,
2021.
ista: Abbatiello A, Bulíček M, Maringová E. 2021. On the dynamic slip boundary condition
for Navier-Stokes-like problems. Mathematical Models and Methods in Applied Sciences.
31(11), 2165–2212.
mla: Abbatiello, Anna, et al. “On the Dynamic Slip Boundary Condition for Navier-Stokes-like
Problems.” Mathematical Models and Methods in Applied Sciences, vol. 31,
no. 11, World Scientific Publishing, 2021, pp. 2165–212, doi:10.1142/S0218202521500470.
short: A. Abbatiello, M. Bulíček, E. Maringová, Mathematical Models and Methods
in Applied Sciences 31 (2021) 2165–2212.
date_created: 2021-12-26T23:01:27Z
date_published: 2021-10-13T00:00:00Z
date_updated: 2025-04-15T08:31:30Z
day: '13'
ddc:
- '510'
department:
- _id: JuFi
doi: 10.1142/S0218202521500470
external_id:
arxiv:
- '2009.09057'
isi:
- '000722309400001'
file:
- access_level: open_access
checksum: 8c0a9396335f0b70e1f5cbfe450a987a
content_type: application/pdf
creator: dernst
date_created: 2022-05-16T10:55:45Z
date_updated: 2022-05-16T10:55:45Z
file_id: '11385'
file_name: 2021_MathModelsMethods_Abbatiello.pdf
file_size: 795483
relation: main_file
success: 1
file_date_updated: 2022-05-16T10:55:45Z
has_accepted_license: '1'
intvolume: ' 31'
isi: 1
issue: '11'
language:
- iso: eng
month: '10'
oa: 1
oa_version: Published Version
page: 2165-2212
project:
- _id: fc31cba2-9c52-11eb-aca3-ff467d239cd2
grant_number: F6504
name: Taming Complexity in Partial Differential Systems
- _id: 260788DE-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: W1245
name: Dissipation and dispersion in nonlinear partial differential equations
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 the dynamic slip boundary condition for Navier-Stokes-like problems
tmp:
image: /images/cc_by_nc_nd.png
legal_code_url: https://creativecommons.org/licenses/by-nc-nd/4.0/legalcode
name: Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International
(CC BY-NC-ND 4.0)
short: CC BY-NC-ND (4.0)
type: journal_article
user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
volume: 31
year: '2021'
...
---
_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: 2025-05-14T10:50:14Z
day: '25'
department:
- _id: JuFi
doi: 10.1142/S0218202521500457
external_id:
arxiv:
- '2009.06917'
isi:
- '000722222900004'
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'
...