---
_id: '10550'
abstract:
- lang: eng
text: The global existence of renormalised solutions and convergence to equilibrium
for reaction-diffusion systems with non-linear diffusion are investigated. The
system is assumed to have quasi-positive non-linearities and to satisfy an entropy
inequality. The difficulties in establishing global renormalised solutions caused
by possibly degenerate diffusion are overcome by introducing a new class of weighted
truncation functions. By means of the obtained global renormalised solutions,
we study the large-time behaviour of complex balanced systems arising from chemical
reaction network theory with non-linear diffusion. When the reaction network does
not admit boundary equilibria, the complex balanced equilibrium is shown, by using
the entropy method, to exponentially attract all renormalised solutions in the
same compatibility class. This convergence extends even to a range of non-linear
diffusion, where global existence is an open problem, yet we are able to show
that solutions to approximate systems converge exponentially to equilibrium uniformly
in the regularisation parameter.
acknowledgement: "We thank the referees for their valuable comments and suggestions.
A major part of this work was carried out when B. Q. Tang visited the Institute
of Science and Technology Austria (ISTA). The hospitality of ISTA is greatly acknowledged.
This work was partially supported by NAWI Graz.\r\nOpen access funding provided
by University of Graz."
article_number: '66'
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Klemens
full_name: Fellner, Klemens
last_name: Fellner
- 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: Michael
full_name: Kniely, Michael
id: 2CA2C08C-F248-11E8-B48F-1D18A9856A87
last_name: Kniely
orcid: 0000-0001-5645-4333
- first_name: Bao Quoc
full_name: Tang, Bao Quoc
last_name: Tang
citation:
ama: Fellner K, Fischer JL, Kniely M, Tang BQ. Global renormalised solutions and
equilibration of reaction-diffusion systems with non-linear diffusion. Journal
of Nonlinear Science. 2023;33. doi:10.1007/s00332-023-09926-w
apa: Fellner, K., Fischer, J. L., Kniely, M., & Tang, B. Q. (2023). Global renormalised
solutions and equilibration of reaction-diffusion systems with non-linear diffusion.
Journal of Nonlinear Science. Springer Nature. https://doi.org/10.1007/s00332-023-09926-w
chicago: Fellner, Klemens, Julian L Fischer, Michael Kniely, and Bao Quoc Tang.
“Global Renormalised Solutions and Equilibration of Reaction-Diffusion Systems
with Non-Linear Diffusion.” Journal of Nonlinear Science. Springer Nature,
2023. https://doi.org/10.1007/s00332-023-09926-w.
ieee: K. Fellner, J. L. Fischer, M. Kniely, and B. Q. Tang, “Global renormalised
solutions and equilibration of reaction-diffusion systems with non-linear diffusion,”
Journal of Nonlinear Science, vol. 33. Springer Nature, 2023.
ista: Fellner K, Fischer JL, Kniely M, Tang BQ. 2023. Global renormalised solutions
and equilibration of reaction-diffusion systems with non-linear diffusion. Journal
of Nonlinear Science. 33, 66.
mla: Fellner, Klemens, et al. “Global Renormalised Solutions and Equilibration of
Reaction-Diffusion Systems with Non-Linear Diffusion.” Journal of Nonlinear
Science, vol. 33, 66, Springer Nature, 2023, doi:10.1007/s00332-023-09926-w.
short: K. Fellner, J.L. Fischer, M. Kniely, B.Q. Tang, Journal of Nonlinear Science
33 (2023).
date_created: 2021-12-16T12:15:35Z
date_published: 2023-06-07T00:00:00Z
date_updated: 2023-08-01T14:40:33Z
day: '07'
ddc:
- '510'
department:
- _id: JuFi
doi: 10.1007/s00332-023-09926-w
external_id:
arxiv:
- '2109.12019'
isi:
- '001002343400002'
file:
- access_level: open_access
checksum: f3f0f0886098e31c81116cff8183750b
content_type: application/pdf
creator: dernst
date_created: 2023-06-19T07:33:53Z
date_updated: 2023-06-19T07:33:53Z
file_id: '13149'
file_name: 2023_JourNonlinearScience_Fellner.pdf
file_size: 742315
relation: main_file
success: 1
file_date_updated: 2023-06-19T07:33:53Z
has_accepted_license: '1'
intvolume: ' 33'
isi: 1
language:
- iso: eng
month: '06'
oa: 1
oa_version: Published Version
publication: Journal of Nonlinear Science
publication_identifier:
eissn:
- 1432-1467
issn:
- 0938-8974
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: Global renormalised solutions and equilibration of reaction-diffusion systems
with non-linear diffusion
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: 33
year: '2023'
...
---
_id: '10547'
abstract:
- lang: eng
text: "We establish global-in-time existence results for thermodynamically consistent
reaction-(cross-)diffusion systems coupled to an equation describing heat transfer.
Our main interest is to model species-dependent diffusivities,\r\nwhile at the
same time ensuring thermodynamic consistency. A key difficulty of the non-isothermal
case lies in the intrinsic presence of cross-diffusion type phenomena like the
Soret and the Dufour effect: due to the temperature/energy dependence of the thermodynamic
equilibria, a nonvanishing temperature gradient may drive a concentration flux
even in a situation with constant concentrations; likewise, a nonvanishing concentration
gradient may drive a heat flux even in a case of spatially constant temperature.
We use time discretisation and regularisation techniques and derive a priori estimates
based on a suitable entropy and the associated entropy production. Renormalised
solutions are used in cases where non-integrable diffusion fluxes or reaction
terms appear."
acknowledgement: M.K. gratefully acknowledges the hospitality of WIAS Berlin, where
a major part of the project was carried out. The research stay of M.K. at WIAS Berlin
was funded by the Austrian Federal Ministry of Education, Science and Research through
a research fellowship for graduates of a promotio sub auspiciis. The research of
A.M. has been partially supported by Deutsche Forschungsgemeinschaft (DFG) through
the Collaborative Research Center SFB 1114 “Scaling Cascades in Complex Systems”
(Project no. 235221301), Subproject C05 “Effective models for materials and interfaces
with multiple scales”. J.F. and A.M. are grateful for the hospitality of the Erwin
Schrödinger Institute in Vienna, where some ideas for this work have been developed.
The authors are grateful to two anonymous referees for several helpful comments,
in particular for the short proof of estimate (2.7).
article_processing_charge: No
article_type: original
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: Katharina
full_name: Hopf, Katharina
last_name: Hopf
- first_name: Michael
full_name: Kniely, Michael
id: 2CA2C08C-F248-11E8-B48F-1D18A9856A87
last_name: Kniely
orcid: 0000-0001-5645-4333
- first_name: Alexander
full_name: Mielke, Alexander
last_name: Mielke
citation:
ama: Fischer JL, Hopf K, Kniely M, Mielke A. Global existence analysis of energy-reaction-diffusion
systems. SIAM Journal on Mathematical Analysis. 2022;54(1):220-267. doi:10.1137/20M1387237
apa: Fischer, J. L., Hopf, K., Kniely, M., & Mielke, A. (2022). Global existence
analysis of energy-reaction-diffusion systems. SIAM Journal on Mathematical
Analysis. Society for Industrial and Applied Mathematics. https://doi.org/10.1137/20M1387237
chicago: Fischer, Julian L, Katharina Hopf, Michael Kniely, and Alexander Mielke.
“Global Existence Analysis of Energy-Reaction-Diffusion Systems.” SIAM Journal
on Mathematical Analysis. Society for Industrial and Applied Mathematics,
2022. https://doi.org/10.1137/20M1387237.
ieee: J. L. Fischer, K. Hopf, M. Kniely, and A. Mielke, “Global existence analysis
of energy-reaction-diffusion systems,” SIAM Journal on Mathematical Analysis,
vol. 54, no. 1. Society for Industrial and Applied Mathematics, pp. 220–267, 2022.
ista: Fischer JL, Hopf K, Kniely M, Mielke A. 2022. Global existence analysis of
energy-reaction-diffusion systems. SIAM Journal on Mathematical Analysis. 54(1),
220–267.
mla: Fischer, Julian L., et al. “Global Existence Analysis of Energy-Reaction-Diffusion
Systems.” SIAM Journal on Mathematical Analysis, vol. 54, no. 1, Society
for Industrial and Applied Mathematics, 2022, pp. 220–67, doi:10.1137/20M1387237.
short: J.L. Fischer, K. Hopf, M. Kniely, A. Mielke, SIAM Journal on Mathematical
Analysis 54 (2022) 220–267.
date_created: 2021-12-16T12:08:56Z
date_published: 2022-01-04T00:00:00Z
date_updated: 2023-08-02T13:37:03Z
day: '04'
department:
- _id: JuFi
doi: 10.1137/20M1387237
external_id:
arxiv:
- '2012.03792 '
isi:
- '000762768000006'
intvolume: ' 54'
isi: 1
issue: '1'
keyword:
- Energy-Reaction-Diffusion Systems
- Cross Diffusion
- Global-In-Time Existence of Weak/Renormalised Solutions
- Entropy Method
- Onsager System
- Soret/Dufour Effect
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://arxiv.org/abs/2012.03792
month: '01'
oa: 1
oa_version: Preprint
page: 220-267
publication: SIAM Journal on Mathematical Analysis
publication_identifier:
issn:
- 0036-1410
publication_status: published
publisher: Society for Industrial and Applied Mathematics
quality_controlled: '1'
scopus_import: '1'
status: public
title: Global existence analysis of energy-reaction-diffusion systems
type: journal_article
user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
volume: 54
year: '2022'
...
---
_id: '7866'
abstract:
- lang: eng
text: In this paper, we establish convergence to equilibrium for a drift–diffusion–recombination
system modelling the charge transport within certain semiconductor devices. More
precisely, we consider a two-level system for electrons and holes which is augmented
by an intermediate energy level for electrons in so-called trapped states. The
recombination dynamics use the mass action principle by taking into account this
additional trap level. The main part of the paper is concerned with the derivation
of an entropy–entropy production inequality, which entails exponential convergence
to the equilibrium via the so-called entropy method. The novelty of our approach
lies in the fact that the entropy method is applied uniformly in a fast-reaction
parameter which governs the lifetime of electrons on the trap level. Thus, the
resulting decay estimate for the densities of electrons and holes extends to the
corresponding quasi-steady-state approximation.
acknowledgement: Open access funding provided by Austrian Science Fund (FWF). The
second author has been supported by the International Research Training Group IGDK
1754 “Optimization and Numerical Analysis for Partial Differential Equations with
Nonsmooth Structures”, funded by the German Research Council (DFG) and the Austrian
Science Fund (FWF) under grant number [W 1244-N18].
article_processing_charge: No
article_type: original
author:
- first_name: Klemens
full_name: Fellner, Klemens
last_name: Fellner
- first_name: Michael
full_name: Kniely, Michael
id: 2CA2C08C-F248-11E8-B48F-1D18A9856A87
last_name: Kniely
orcid: 0000-0001-5645-4333
citation:
ama: Fellner K, Kniely M. Uniform convergence to equilibrium for a family of drift–diffusion
models with trap-assisted recombination and the limiting Shockley–Read–Hall model.
Journal of Elliptic and Parabolic Equations. 2020;6:529-598. doi:10.1007/s41808-020-00068-8
apa: Fellner, K., & Kniely, M. (2020). Uniform convergence to equilibrium for
a family of drift–diffusion models with trap-assisted recombination and the limiting
Shockley–Read–Hall model. Journal of Elliptic and Parabolic Equations.
Springer Nature. https://doi.org/10.1007/s41808-020-00068-8
chicago: Fellner, Klemens, and Michael Kniely. “Uniform Convergence to Equilibrium
for a Family of Drift–Diffusion Models with Trap-Assisted Recombination and the
Limiting Shockley–Read–Hall Model.” Journal of Elliptic and Parabolic Equations.
Springer Nature, 2020. https://doi.org/10.1007/s41808-020-00068-8.
ieee: K. Fellner and M. Kniely, “Uniform convergence to equilibrium for a family
of drift–diffusion models with trap-assisted recombination and the limiting Shockley–Read–Hall
model,” Journal of Elliptic and Parabolic Equations, vol. 6. Springer Nature,
pp. 529–598, 2020.
ista: Fellner K, Kniely M. 2020. Uniform convergence to equilibrium for a family
of drift–diffusion models with trap-assisted recombination and the limiting Shockley–Read–Hall
model. Journal of Elliptic and Parabolic Equations. 6, 529–598.
mla: Fellner, Klemens, and Michael Kniely. “Uniform Convergence to Equilibrium for
a Family of Drift–Diffusion Models with Trap-Assisted Recombination and the Limiting
Shockley–Read–Hall Model.” Journal of Elliptic and Parabolic Equations,
vol. 6, Springer Nature, 2020, pp. 529–98, doi:10.1007/s41808-020-00068-8.
short: K. Fellner, M. Kniely, Journal of Elliptic and Parabolic Equations 6 (2020)
529–598.
date_created: 2020-05-17T22:00:45Z
date_published: 2020-12-01T00:00:00Z
date_updated: 2021-01-12T08:15:47Z
day: '01'
ddc:
- '510'
department:
- _id: JuFi
doi: 10.1007/s41808-020-00068-8
file:
- access_level: open_access
checksum: 6bc6832caacddceee1471291e93dcf1d
content_type: application/pdf
creator: dernst
date_created: 2020-11-25T08:59:59Z
date_updated: 2020-11-25T08:59:59Z
file_id: '8802'
file_name: 2020_JourEllipticParabEquat_Fellner.pdf
file_size: 8408694
relation: main_file
success: 1
file_date_updated: 2020-11-25T08:59:59Z
has_accepted_license: '1'
intvolume: ' 6'
language:
- iso: eng
month: '12'
oa: 1
oa_version: Published Version
page: 529-598
project:
- _id: 3AC91DDA-15DF-11EA-824D-93A3E7B544D1
call_identifier: FWF
name: FWF Open Access Fund
publication: Journal of Elliptic and Parabolic Equations
publication_identifier:
eissn:
- '22969039'
issn:
- '22969020'
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: Uniform convergence to equilibrium for a family of drift–diffusion models with
trap-assisted recombination and the limiting Shockley–Read–Hall model
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: 6
year: '2020'
...
---
_id: '8697'
abstract:
- lang: eng
text: In the computation of the material properties of random alloys, the method
of 'special quasirandom structures' attempts to approximate the properties of
the alloy on a finite volume with higher accuracy by replicating certain statistics
of the random atomic lattice in the finite volume as accurately as possible. In
the present work, we provide a rigorous justification for a variant of this method
in the framework of the Thomas–Fermi–von Weizsäcker (TFW) model. Our approach
is based on a recent analysis of a related variance reduction method in stochastic
homogenization of linear elliptic PDEs and the locality properties of the TFW
model. Concerning the latter, we extend an exponential locality result by Nazar
and Ortner to include point charges, a result that may be of independent interest.
article_processing_charge: Yes (via OA deal)
article_type: original
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: Michael
full_name: Kniely, Michael
id: 2CA2C08C-F248-11E8-B48F-1D18A9856A87
last_name: Kniely
orcid: 0000-0001-5645-4333
citation:
ama: Fischer JL, Kniely M. Variance reduction for effective energies of random lattices
in the Thomas-Fermi-von Weizsäcker model. Nonlinearity. 2020;33(11):5733-5772.
doi:10.1088/1361-6544/ab9728
apa: Fischer, J. L., & Kniely, M. (2020). Variance reduction for effective energies
of random lattices in the Thomas-Fermi-von Weizsäcker model. Nonlinearity.
IOP Publishing. https://doi.org/10.1088/1361-6544/ab9728
chicago: Fischer, Julian L, and Michael Kniely. “Variance Reduction for Effective
Energies of Random Lattices in the Thomas-Fermi-von Weizsäcker Model.” Nonlinearity.
IOP Publishing, 2020. https://doi.org/10.1088/1361-6544/ab9728.
ieee: J. L. Fischer and M. Kniely, “Variance reduction for effective energies of
random lattices in the Thomas-Fermi-von Weizsäcker model,” Nonlinearity,
vol. 33, no. 11. IOP Publishing, pp. 5733–5772, 2020.
ista: Fischer JL, Kniely M. 2020. Variance reduction for effective energies of random
lattices in the Thomas-Fermi-von Weizsäcker model. Nonlinearity. 33(11), 5733–5772.
mla: Fischer, Julian L., and Michael Kniely. “Variance Reduction for Effective Energies
of Random Lattices in the Thomas-Fermi-von Weizsäcker Model.” Nonlinearity,
vol. 33, no. 11, IOP Publishing, 2020, pp. 5733–72, doi:10.1088/1361-6544/ab9728.
short: J.L. Fischer, M. Kniely, Nonlinearity 33 (2020) 5733–5772.
date_created: 2020-10-25T23:01:16Z
date_published: 2020-11-01T00:00:00Z
date_updated: 2023-08-22T10:38:38Z
day: '01'
ddc:
- '510'
department:
- _id: JuFi
doi: 10.1088/1361-6544/ab9728
external_id:
arxiv:
- '1906.12245'
isi:
- '000576492700001'
file:
- access_level: open_access
checksum: ed90bc6eb5f32ee6157fef7f3aabc057
content_type: application/pdf
creator: cziletti
date_created: 2020-10-27T12:09:57Z
date_updated: 2020-10-27T12:09:57Z
file_id: '8710'
file_name: 2020_Nonlinearity_Fischer.pdf
file_size: 1223899
relation: main_file
success: 1
file_date_updated: 2020-10-27T12:09:57Z
has_accepted_license: '1'
intvolume: ' 33'
isi: 1
issue: '11'
language:
- iso: eng
license: https://creativecommons.org/licenses/by/3.0/
month: '11'
oa: 1
oa_version: Published Version
page: 5733-5772
publication: Nonlinearity
publication_identifier:
eissn:
- '13616544'
issn:
- '09517715'
publication_status: published
publisher: IOP Publishing
quality_controlled: '1'
scopus_import: '1'
status: public
title: Variance reduction for effective energies of random lattices in the Thomas-Fermi-von
Weizsäcker model
tmp:
image: /images/cc_by.png
legal_code_url: https://creativecommons.org/licenses/by/3.0/legalcode
name: Creative Commons Attribution 3.0 Unported (CC BY 3.0)
short: CC BY (3.0)
type: journal_article
user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
volume: 33
year: '2020'
...
---
_id: '6762'
abstract:
- lang: eng
text: "We present and study novel optimal control problems motivated by the search
for photovoltaic materials with high power-conversion efficiency. The material
must perform the first step: convert light (photons) into electronic excitations.
We formulate various desirable properties of the excitations as mathematical control
goals at the Kohn-Sham-DFT level\r\nof theory, with the control being given by
the nuclear charge distribution. We prove that nuclear distributions exist which
give rise to optimal HOMO-LUMO excitations, and present illustrative numerical
simulations for 1D finite nanocrystals. We observe pronounced goal-dependent features
such as large electron-hole separation, and a hierarchy of length scales: internal
HOMO and LUMO wavelengths < atomic spacings < (irregular) fluctuations of the
doping profiles < system size."
article_processing_charge: No
arxiv: 1
author:
- first_name: Gero
full_name: Friesecke, Gero
last_name: Friesecke
- first_name: Michael
full_name: Kniely, Michael
id: 2CA2C08C-F248-11E8-B48F-1D18A9856A87
last_name: Kniely
orcid: 0000-0001-5645-4333
citation:
ama: Friesecke G, Kniely M. New optimal control problems in density functional theory
motivated by photovoltaics. Multiscale Modeling and Simulation. 2019;17(3):926-947.
doi:10.1137/18M1207272
apa: Friesecke, G., & Kniely, M. (2019). New optimal control problems in density
functional theory motivated by photovoltaics. Multiscale Modeling and Simulation.
SIAM. https://doi.org/10.1137/18M1207272
chicago: Friesecke, Gero, and Michael Kniely. “New Optimal Control Problems in Density
Functional Theory Motivated by Photovoltaics.” Multiscale Modeling and Simulation.
SIAM, 2019. https://doi.org/10.1137/18M1207272.
ieee: G. Friesecke and M. Kniely, “New optimal control problems in density functional
theory motivated by photovoltaics,” Multiscale Modeling and Simulation,
vol. 17, no. 3. SIAM, pp. 926–947, 2019.
ista: Friesecke G, Kniely M. 2019. New optimal control problems in density functional
theory motivated by photovoltaics. Multiscale Modeling and Simulation. 17(3),
926–947.
mla: Friesecke, Gero, and Michael Kniely. “New Optimal Control Problems in Density
Functional Theory Motivated by Photovoltaics.” Multiscale Modeling and Simulation,
vol. 17, no. 3, SIAM, 2019, pp. 926–47, doi:10.1137/18M1207272.
short: G. Friesecke, M. Kniely, Multiscale Modeling and Simulation 17 (2019) 926–947.
date_created: 2019-08-04T21:59:21Z
date_published: 2019-07-16T00:00:00Z
date_updated: 2023-09-05T15:05:45Z
day: '16'
department:
- _id: JuFi
doi: 10.1137/18M1207272
external_id:
arxiv:
- '1808.04200'
isi:
- '000487931800002'
intvolume: ' 17'
isi: 1
issue: '3'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://arxiv.org/abs/1808.04200
month: '07'
oa: 1
oa_version: Preprint
page: 926-947
publication: Multiscale Modeling and Simulation
publication_identifier:
eissn:
- '15403467'
issn:
- '15403459'
publication_status: published
publisher: SIAM
quality_controlled: '1'
scopus_import: '1'
status: public
title: New optimal control problems in density functional theory motivated by photovoltaics
type: journal_article
user_id: c635000d-4b10-11ee-a964-aac5a93f6ac1
volume: 17
year: '2019'
...