---
_id: '10613'
abstract:
- lang: eng
text: Motivated by the recent preprint [\emph{arXiv:2004.08412}] by Ayala, Carinci,
and Redig, we first provide a general framework for the study of scaling limits
of higher-order fields. Then, by considering the same class of infinite interacting
particle systems as in [\emph{arXiv:2004.08412}], namely symmetric simple exclusion
and inclusion processes in the d-dimensional Euclidean lattice, we prove the hydrodynamic
limit, and convergence for the equilibrium fluctuations, of higher-order fields.
In particular, the limit fields exhibit a tensor structure. Our fluctuation result
differs from that in [\emph{arXiv:2004.08412}], since we considered-dimensional
Euclidean lattice, we prove the hydrodynamic limit, and convergence for the equilibrium
fluctuations, of higher-order fields. In particular, the limit fields exhibit
a tensor structure. Our fluctuation result differs from that in [\emph{arXiv:2004.08412}],
since we consider a different notion of higher-order fluctuation fields.
acknowledgement: "F.S. would like to thank Mario Ayala and Frank Redig for useful
discussions. J.P.C. acknowledges partial financial support from the US National
Science Foundation (DMS-1855604). F.S. was financially supported by the European
Union’s Horizon 2020 research and innovation programme under the Marie-Skłodowska-Curie
grant agreement No. 754411.\r\n"
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Joe P.
full_name: Chen, Joe P.
last_name: Chen
- first_name: Federico
full_name: Sau, Federico
id: E1836206-9F16-11E9-8814-AEFDE5697425
last_name: Sau
citation:
ama: Chen JP, Sau F. Higher-order hydrodynamics and equilibrium fluctuations of
interacting particle systems. Markov Processes And Related Fields. 2021;27(3):339-380.
apa: Chen, J. P., & Sau, F. (2021). Higher-order hydrodynamics and equilibrium
fluctuations of interacting particle systems. Markov Processes And Related
Fields. Polymat Publishing.
chicago: Chen, Joe P., and Federico Sau. “Higher-Order Hydrodynamics and Equilibrium
Fluctuations of Interacting Particle Systems.” Markov Processes And Related
Fields. Polymat Publishing, 2021.
ieee: J. P. Chen and F. Sau, “Higher-order hydrodynamics and equilibrium fluctuations
of interacting particle systems,” Markov Processes And Related Fields,
vol. 27, no. 3. Polymat Publishing, pp. 339–380, 2021.
ista: Chen JP, Sau F. 2021. Higher-order hydrodynamics and equilibrium fluctuations
of interacting particle systems. Markov Processes And Related Fields. 27(3), 339–380.
mla: Chen, Joe P., and Federico Sau. “Higher-Order Hydrodynamics and Equilibrium
Fluctuations of Interacting Particle Systems.” Markov Processes And Related
Fields, vol. 27, no. 3, Polymat Publishing, 2021, pp. 339–80.
short: J.P. Chen, F. Sau, Markov Processes And Related Fields 27 (2021) 339–380.
corr_author: '1'
date_created: 2022-01-10T14:02:31Z
date_published: 2021-03-16T00:00:00Z
date_updated: 2025-06-26T11:57:53Z
day: '16'
department:
- _id: JaMa
ec_funded: 1
external_id:
arxiv:
- '2008.13403'
intvolume: ' 27'
issue: '3'
keyword:
- interacting particle systems
- higher-order fields
- hydrodynamic limit
- equilibrium fluctuations
- duality
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://arxiv.org/abs/2008.13403
month: '03'
oa: 1
oa_version: Preprint
page: 339-380
project:
- _id: 260C2330-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '754411'
name: ISTplus - Postdoctoral Fellowships
publication: Markov Processes And Related Fields
publication_identifier:
issn:
- 1024-2953
publication_status: published
publisher: Polymat Publishing
quality_controlled: '1'
related_material:
link:
- description: Link to Abstract on publisher's website
relation: other
url: http://math-mprf.org/journal/articles/id1614/
- description: Referred to in Abstract
relation: used_for_analysis_in
url: https://arxiv.org/abs/2004.08412
status: public
title: Higher-order hydrodynamics and equilibrium fluctuations of interacting particle
systems
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 27
year: '2021'
...
---
OA_type: closed access
_id: '1317'
abstract:
- lang: eng
text: We analyze the behaviour of free boundaries in thin-film flow in the regime
of strong slippage n∈[1,2) and in the regime of very weak slippage n∈,3) qualitatively
and quantitatively. In the regime of strong slippage, we construct initial data
which are bounded from above by the steady state but for which nevertheless instantaneous
forward motion of the free boundary occurs. This shows that the initial behaviour
of the free boundary is not determined just by the growth of the initial data
at the free boundary. Note that this is a new phenomenon for degenerate parabolic
equations which is specific for higher-order equations. Furthermore, this result
resolves a controversy in the literature over optimality of sufficient conditions
for the occurrence of a waiting time phenomenon. In contrast, in the regime of
very weak slippage we derive lower bounds on free boundary propagation which are
optimal in the sense that they coincide up to a constant factor with the known
upper bounds. In particular, in this regime the growth of the initial data at
the free boundary fully determines the initial behaviour of the interface.
article_processing_charge: No
article_type: original
author:
- first_name: Julian L
full_name: Fischer, Julian L
id: 2C12A0B0-F248-11E8-B48F-1D18A9856A87
last_name: Fischer
orcid: 0000-0002-0479-558X
citation:
ama: 'Fischer JL. Behaviour of free boundaries in thin-film flow: The regime of
strong slippage and the regime of very weak slippage. Annales de l’Institut
Henri Poincare (C) Non Linear Analysis. 2016;33(5):1301-1327. doi:10.1016/j.anihpc.2015.05.001'
apa: 'Fischer, J. L. (2016). Behaviour of free boundaries in thin-film flow: The
regime of strong slippage and the regime of very weak slippage. Annales de
l’Institut Henri Poincare (C) Non Linear Analysis. EMS Press. https://doi.org/10.1016/j.anihpc.2015.05.001'
chicago: 'Fischer, Julian L. “Behaviour of Free Boundaries in Thin-Film Flow: The
Regime of Strong Slippage and the Regime of Very Weak Slippage.” Annales de
l’Institut Henri Poincare (C) Non Linear Analysis. EMS Press, 2016. https://doi.org/10.1016/j.anihpc.2015.05.001.'
ieee: 'J. L. Fischer, “Behaviour of free boundaries in thin-film flow: The regime
of strong slippage and the regime of very weak slippage,” Annales de l’Institut
Henri Poincare (C) Non Linear Analysis, vol. 33, no. 5. EMS Press, pp. 1301–1327,
2016.'
ista: 'Fischer JL. 2016. Behaviour of free boundaries in thin-film flow: The regime
of strong slippage and the regime of very weak slippage. Annales de l’Institut
Henri Poincare (C) Non Linear Analysis. 33(5), 1301–1327.'
mla: 'Fischer, Julian L. “Behaviour of Free Boundaries in Thin-Film Flow: The Regime
of Strong Slippage and the Regime of Very Weak Slippage.” Annales de l’Institut
Henri Poincare (C) Non Linear Analysis, vol. 33, no. 5, EMS Press, 2016, pp.
1301–27, doi:10.1016/j.anihpc.2015.05.001.'
short: J.L. Fischer, Annales de l’Institut Henri Poincare (C) Non Linear Analysis
33 (2016) 1301–1327.
date_created: 2018-12-11T11:51:20Z
date_published: 2016-10-01T00:00:00Z
date_updated: 2026-05-06T07:02:19Z
day: '01'
doi: 10.1016/j.anihpc.2015.05.001
extern: '1'
fulldoi: https://doi.org/10.1016/j.anihpc.2015.05.001
intvolume: ' 33'
issue: '5'
keyword:
- Thin-film equation
- Free boundary
- Waiting time
- Qualitative behaviour
- Higher-order parabolic equation
- Degenerate parabolic equation
language:
- iso: eng
month: '10'
oa_version: None
page: 1301 - 1327
publication: Annales de l'Institut Henri Poincare (C) Non Linear Analysis
publication_identifier:
eissn:
- 1873-1430
issnl:
- 0294-1449
publication_status: published
publisher: EMS Press
publist_id: '5952'
quality_controlled: '1'
status: public
title: 'Behaviour of free boundaries in thin-film flow: The regime of strong slippage
and the regime of very weak slippage'
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 33
year: '2016'
...
---
OA_place: publisher
OA_type: gold
_id: '1318'
abstract:
- lang: eng
text: 'We develop a large-scale regularity theory of higher order for divergence-form
elliptic equations with heterogeneous coefficient fields a in the context of stochastic
homogenization. The large-scale regularity of a-harmonic functions is encoded
by Liouville principles: The space of a-harmonic functions that grow at most like
a polynomial of degree k has the same dimension as in the constant-coefficient
case. This result can be seen as the qualitative side of a large-scale Ck,α-regularity
theory, which in the present work is developed in the form of a corresponding
Ck,α-“excess decay” estimate: For a given a-harmonic function u on a ball BR,
its energy distance on some ball Br to the above space of a-harmonic functions
that grow at most like a polynomial of degree k has the natural decay in the radius
r above some minimal radius r0. Though motivated by stochastic homogenization,
the contribution of this paper is of purely deterministic nature: We work under
the assumption that for the given realization a of the coefficient field, the
couple (φ, σ) of scalar and vector potentials of the harmonic coordinates, where
φ is the usual corrector, grows sublinearly in a mildly quantified way. We then
construct “kth-order correctors” and thereby the space of a-harmonic functions
that grow at most like a polynomial of degree k, establish the above excess decay,
and then the corresponding Liouville principle.'
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: Felix
full_name: Otto, Felix
last_name: Otto
citation:
ama: Fischer JL, Otto F. A higher-order large scale regularity theory for random
elliptic operators. Communications in Partial Differential Equations. 2016;41(7):1108-1148.
doi:10.1080/03605302.2016.1179318
apa: Fischer, J. L., & Otto, F. (2016). A higher-order large scale regularity
theory for random elliptic operators. Communications in Partial Differential
Equations. Taylor & Francis. https://doi.org/10.1080/03605302.2016.1179318
chicago: Fischer, Julian L, and Felix Otto. “A Higher-Order Large Scale Regularity
Theory for Random Elliptic Operators.” Communications in Partial Differential
Equations. Taylor & Francis, 2016. https://doi.org/10.1080/03605302.2016.1179318.
ieee: J. L. Fischer and F. Otto, “A higher-order large scale regularity theory for
random elliptic operators,” Communications in Partial Differential Equations,
vol. 41, no. 7. Taylor & Francis, pp. 1108–1148, 2016.
ista: Fischer JL, Otto F. 2016. A higher-order large scale regularity theory for
random elliptic operators. Communications in Partial Differential Equations. 41(7),
1108–1148.
mla: Fischer, Julian L., and Felix Otto. “A Higher-Order Large Scale Regularity
Theory for Random Elliptic Operators.” Communications in Partial Differential
Equations, vol. 41, no. 7, Taylor & Francis, 2016, pp. 1108–48, doi:10.1080/03605302.2016.1179318.
short: J.L. Fischer, F. Otto, Communications in Partial Differential Equations 41
(2016) 1108–1148.
date_created: 2018-12-11T11:51:20Z
date_published: 2016-06-16T00:00:00Z
date_updated: 2026-05-06T07:05:16Z
day: '16'
doi: 10.1080/03605302.2016.1179318
extern: '1'
external_id:
arxiv:
- '1503.07578'
fulldoi: https://doi.org/10.1080/03605302.2016.1179318
intvolume: ' 41'
issue: '7'
keyword:
- Ck
- α regularity
- higher-ordercorrectors
- Liouville principle
- random elliptic operator
- regularity theory
- stochastic homogenization
language:
- iso: eng
license: https://creativecommons.org/licenses/by/4.0/
main_file_link:
- open_access: '1'
url: https://doi.org/10.1080/03605302.2016.1179318
month: '06'
oa: 1
oa_version: Published Version
page: 1108 - 1148
publication: Communications in Partial Differential Equations
publication_identifier:
eissn:
- 1532-4133
issnl:
- 0360-5302
publication_status: published
publisher: Taylor & Francis
publist_id: '5953'
quality_controlled: '1'
status: public
title: A higher-order large scale regularity theory for random elliptic operators
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: 41
year: '2016'
...