---
_id: '12304'
abstract:
- lang: eng
text: 'We establish sharp criteria for the instantaneous propagation of free boundaries
in solutions to the thin-film equation. The criteria are formulated in terms of
the initial distribution of mass (as opposed to previous almost-optimal results),
reflecting the fact that mass is a locally conserved quantity for the thin-film
equation. In the regime of weak slippage, our criteria are at the same time necessary
and sufficient. The proof of our upper bounds on free boundary propagation is
based on a strategy of “propagation of degeneracy” down to arbitrarily small spatial
scales: We combine estimates on the local mass and estimates on energies to show
that “degeneracy” on a certain space-time cylinder entails “degeneracy” on a spatially
smaller space-time cylinder with the same time horizon. The derivation of our
lower bounds on free boundary propagation is based on a combination of a monotone
quantity and almost optimal estimates established previously by the second author
with a new estimate connecting motion of mass to entropy production.'
acknowledgement: N. De Nitti acknowledges the kind hospitality of IST Austria within
the framework of the ISTernship Summer Program 2018, during which most of the present
article was written. N. DeNitti has received funding by The Austrian Agency for
International Cooperation in Education &Research (OeAD-GmbH) via its financial support
of the ISTernship Summer Program 2018. N.De Nitti would also like to thank Giuseppe
Coclite, Giuseppe Devillanova, Giuseppe Florio, Sebastian Hensel, and Francesco
Maddalena for several helpful conversations on topics related to this work.
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Nicola
full_name: De Nitti, Nicola
last_name: De Nitti
- 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: De Nitti N, Fischer JL. Sharp criteria for the waiting time phenomenon in solutions
to the thin-film equation. Communications in Partial Differential Equations.
2022;47(7):1394-1434. doi:10.1080/03605302.2022.2056702
apa: De Nitti, N., & Fischer, J. L. (2022). Sharp criteria for the waiting time
phenomenon in solutions to the thin-film equation. Communications in Partial
Differential Equations. Taylor & Francis. https://doi.org/10.1080/03605302.2022.2056702
chicago: De Nitti, Nicola, and Julian L Fischer. “Sharp Criteria for the Waiting
Time Phenomenon in Solutions to the Thin-Film Equation.” Communications in
Partial Differential Equations. Taylor & Francis, 2022. https://doi.org/10.1080/03605302.2022.2056702.
ieee: N. De Nitti and J. L. Fischer, “Sharp criteria for the waiting time phenomenon
in solutions to the thin-film equation,” Communications in Partial Differential
Equations, vol. 47, no. 7. Taylor & Francis, pp. 1394–1434, 2022.
ista: De Nitti N, Fischer JL. 2022. Sharp criteria for the waiting time phenomenon
in solutions to the thin-film equation. Communications in Partial Differential
Equations. 47(7), 1394–1434.
mla: De Nitti, Nicola, and Julian L. Fischer. “Sharp Criteria for the Waiting Time
Phenomenon in Solutions to the Thin-Film Equation.” Communications in Partial
Differential Equations, vol. 47, no. 7, Taylor & Francis, 2022, pp. 1394–434,
doi:10.1080/03605302.2022.2056702.
short: N. De Nitti, J.L. Fischer, Communications in Partial Differential Equations
47 (2022) 1394–1434.
corr_author: '1'
date_created: 2023-01-16T10:06:50Z
date_published: 2022-07-01T00:00:00Z
date_updated: 2024-10-09T21:03:57Z
day: '01'
department:
- _id: JuFi
doi: 10.1080/03605302.2022.2056702
external_id:
arxiv:
- '1907.05342'
isi:
- '000805689800001'
intvolume: ' 47'
isi: 1
issue: '7'
keyword:
- Applied Mathematics
- Analysis
language:
- iso: eng
main_file_link:
- open_access: '1'
url: ' https://doi.org/10.48550/arXiv.1907.05342'
month: '07'
oa: 1
oa_version: Preprint
page: 1394-1434
publication: Communications in Partial Differential Equations
publication_identifier:
eissn:
- 1532-4133
issn:
- 0360-5302
publication_status: published
publisher: Taylor & Francis
quality_controlled: '1'
scopus_import: '1'
status: public
title: Sharp criteria for the waiting time phenomenon in solutions to the thin-film
equation
type: journal_article
user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
volume: 47
year: '2022'
...
---
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'
intvolume: ' 41'
issue: '7'
keyword:
- Ck
- α regularity
- higher-ordercorrectors
- Liouville principle
- random elliptic operator
- regularity theory
- stochastic homogenization
language:
- iso: eng
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'
...
---
OA_place: repository
OA_type: green
_id: '22047'
abstract:
- lang: eng
text: "We undertake a comprehensive study of the nonlinear Schrödinger equation
(mathematical formular) where u(t, x) is a complex-valued function in spacetime
R, xRn/x, λ1 and λ2 are nonzero real constants, and (mathematical formular). We
address questions related to local and global well-posedness, finite time blowup,
and asymptotic behaviour. Scattering is considered both in the energy space H^1(ℝ
n ) and in the pseudoconformal space Σ := {f ∈ H^1(ℝ^n); xf ∈ L^2(ℝ^n)}. Of particular
interest is the case when both nonlinearities are defocusing and correspond to
the L2/x-critical, respectively H1/x-critical NLS, that is, λ1, λ2 > 0 and (mathematical
formular) . The results at the endpoint p1= 4/n are conditional on a conjectured
global existence and spacetime estimate for the L2/x-critical nonlinear Schrödinger
equation, which has been verified in dimensions n ≥ 2 for radial data in Tao et
al. (Tao et al. to appear a,b) and Killip et al. (preprint).\r\nAs an off-shoot
of our analysis, we also obtain a new, simpler proof of scattering in H1/x for
solutions to the nonlinear Schrödinger equation (mathematical formular) with 4/n
< p < 4/n-2, which was first obtained by Ginibre and Velo (Citation1985)."
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Terence
full_name: Tao, Terence
last_name: Tao
- first_name: Monica
full_name: Visan, Monica
id: 056daca0-b8d1-11f0-964f-f91054abf8ca
last_name: Visan
- first_name: Xiaoyi
full_name: Zhang, Xiaoyi
last_name: Zhang
citation:
ama: Tao T, Vişan M, Zhang X. The nonlinear Schrödinger equation with combined power-type
nonlinearities. Communications in Partial Differential Equations. 2007;32(8):1281-1343.
doi:10.1080/03605300701588805
apa: Tao, T., Vişan, M., & Zhang, X. (2007). The nonlinear Schrödinger equation
with combined power-type nonlinearities. Communications in Partial Differential
Equations. Informa UK Limited. https://doi.org/10.1080/03605300701588805
chicago: Tao, Terence, Monica Vişan, and Xiaoyi Zhang. “The Nonlinear Schrödinger
Equation with Combined Power-Type Nonlinearities.” Communications in Partial
Differential Equations. Informa UK Limited, 2007. https://doi.org/10.1080/03605300701588805.
ieee: T. Tao, M. Vişan, and X. Zhang, “The nonlinear Schrödinger equation with combined
power-type nonlinearities,” Communications in Partial Differential Equations,
vol. 32, no. 8. Informa UK Limited, pp. 1281–1343, 2007.
ista: Tao T, Vişan M, Zhang X. 2007. The nonlinear Schrödinger equation with combined
power-type nonlinearities. Communications in Partial Differential Equations. 32(8),
1281–1343.
mla: Tao, Terence, et al. “The Nonlinear Schrödinger Equation with Combined Power-Type
Nonlinearities.” Communications in Partial Differential Equations, vol.
32, no. 8, Informa UK Limited, 2007, pp. 1281–343, doi:10.1080/03605300701588805.
short: T. Tao, M. Vişan, X. Zhang, Communications in Partial Differential Equations
32 (2007) 1281–1343.
das_tickbox: '1'
date_created: 2026-06-19T07:49:46Z
date_published: 2007-08-29T00:00:00Z
date_updated: 2026-06-25T08:04:20Z
day: '29'
doi: 10.1080/03605300701588805
extern: '1'
external_id:
arxiv:
- math/0511070
intvolume: ' 32'
issue: '8'
keyword:
- Energy-critical
- Mass-critical
- Nonlinear Schrödinger equation
- Wellposedness
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://doi.org/10.48550/arXiv.math/0511070
mathsc:
- 35Q55
month: '08'
oa: 1
oa_version: Preprint
page: 1281-1343
publication: Communications in Partial Differential Equations
publication_identifier:
eissn:
- 1532-4133
issn:
- 0360-5302
publication_status: published
publisher: Informa UK Limited
quality_controlled: '1'
scopus_import: '1'
status: public
title: The nonlinear Schrödinger equation with combined power-type nonlinearities
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 32
year: '2007'
...