---
_id: '11701'
abstract:
- lang: eng
text: In this paper we develop a new approach to nonlinear stochastic partial differential
equations with Gaussian noise. Our aim is to provide an abstract framework which
is applicable to a large class of SPDEs and includes many important cases of nonlinear
parabolic problems which are of quasi- or semilinear type. This first part is
on local existence and well-posedness. A second part in preparation is on blow-up
criteria and regularization. Our theory is formulated in an Lp-setting, and because
of this we can deal with nonlinearities in a very efficient way. Applications
to several concrete problems and their quasilinear variants are given. This includes
Burgers' equation, the Allen–Cahn equation, the Cahn–Hilliard equation, reaction–diffusion
equations, and the porous media equation. The interplay of the nonlinearities
and the critical spaces of initial data leads to new results and insights for
these SPDEs. The proofs are based on recent developments in maximal regularity
theory for the linearized problem for deterministic and stochastic evolution equations.
In particular, our theory can be seen as a stochastic version of the theory of
critical spaces due to Prüss–Simonett–Wilke (2018). Sharp weighted time-regularity
allow us to deal with rough initial values and obtain instantaneous regularization
results. The abstract well-posedness results are obtained by a combination of
several sophisticated splitting and truncation arguments.
acknowledgement: The second author is supported by the VIDI subsidy 639.032.427 of
the Netherlands Organisation for Scientific Research (NWO).
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Antonio
full_name: Agresti, Antonio
id: 673cd0cc-9b9a-11eb-b144-88f30e1fbb72
last_name: Agresti
orcid: 0000-0002-9573-2962
- first_name: Mark
full_name: Veraar, Mark
last_name: Veraar
citation:
ama: Agresti A, Veraar M. Nonlinear parabolic stochastic evolution equations in
critical spaces Part I. Stochastic maximal regularity and local existence. Nonlinearity.
2022;35(8):4100-4210. doi:10.1088/1361-6544/abd613
apa: Agresti, A., & Veraar, M. (2022). Nonlinear parabolic stochastic evolution
equations in critical spaces Part I. Stochastic maximal regularity and local existence.
Nonlinearity. IOP Publishing. https://doi.org/10.1088/1361-6544/abd613
chicago: Agresti, Antonio, and Mark Veraar. “Nonlinear Parabolic Stochastic Evolution
Equations in Critical Spaces Part I. Stochastic Maximal Regularity and Local Existence.”
Nonlinearity. IOP Publishing, 2022. https://doi.org/10.1088/1361-6544/abd613.
ieee: A. Agresti and M. Veraar, “Nonlinear parabolic stochastic evolution equations
in critical spaces Part I. Stochastic maximal regularity and local existence,”
Nonlinearity, vol. 35, no. 8. IOP Publishing, pp. 4100–4210, 2022.
ista: Agresti A, Veraar M. 2022. Nonlinear parabolic stochastic evolution equations
in critical spaces Part I. Stochastic maximal regularity and local existence.
Nonlinearity. 35(8), 4100–4210.
mla: Agresti, Antonio, and Mark Veraar. “Nonlinear Parabolic Stochastic Evolution
Equations in Critical Spaces Part I. Stochastic Maximal Regularity and Local Existence.”
Nonlinearity, vol. 35, no. 8, IOP Publishing, 2022, pp. 4100–210, doi:10.1088/1361-6544/abd613.
short: A. Agresti, M. Veraar, Nonlinearity 35 (2022) 4100–4210.
date_created: 2022-07-31T22:01:47Z
date_published: 2022-08-04T00:00:00Z
date_updated: 2023-08-03T12:25:08Z
day: '04'
ddc:
- '510'
department:
- _id: JuFi
doi: 10.1088/1361-6544/abd613
external_id:
arxiv:
- '2001.00512'
isi:
- '000826695900001'
file:
- access_level: open_access
checksum: 997a4bff2dfbee3321d081328c2f1e1a
content_type: application/pdf
creator: dernst
date_created: 2022-08-01T10:39:36Z
date_updated: 2022-08-01T10:39:36Z
file_id: '11715'
file_name: 2022_Nonlinearity_Agresti.pdf
file_size: 2122096
relation: main_file
success: 1
file_date_updated: 2022-08-01T10:39:36Z
has_accepted_license: '1'
intvolume: ' 35'
isi: 1
issue: '8'
language:
- iso: eng
month: '08'
oa: 1
oa_version: Published Version
page: 4100-4210
publication: Nonlinearity
publication_identifier:
eissn:
- 1361-6544
issn:
- 0951-7715
publication_status: published
publisher: IOP Publishing
quality_controlled: '1'
scopus_import: '1'
status: public
title: Nonlinear parabolic stochastic evolution equations in critical spaces Part
I. Stochastic maximal regularity and local existence
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: 35
year: '2022'
...
---
_id: '8420'
abstract:
- lang: eng
text: We show that in the space of all convex billiard boundaries, the set of boundaries
with rational caustics is dense. More precisely, the set of billiard boundaries
with caustics of rotation number 1/q is polynomially sense in the smooth case,
and exponentially dense in the analytic case.
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Vadim
full_name: Kaloshin, Vadim
id: FE553552-CDE8-11E9-B324-C0EBE5697425
last_name: Kaloshin
orcid: 0000-0002-6051-2628
- first_name: Ke
full_name: Zhang, Ke
last_name: Zhang
citation:
ama: Kaloshin V, Zhang K. Density of convex billiards with rational caustics. Nonlinearity.
2018;31(11):5214-5234. doi:10.1088/1361-6544/aadc12
apa: Kaloshin, V., & Zhang, K. (2018). Density of convex billiards with rational
caustics. Nonlinearity. IOP Publishing. https://doi.org/10.1088/1361-6544/aadc12
chicago: Kaloshin, Vadim, and Ke Zhang. “Density of Convex Billiards with Rational
Caustics.” Nonlinearity. IOP Publishing, 2018. https://doi.org/10.1088/1361-6544/aadc12.
ieee: V. Kaloshin and K. Zhang, “Density of convex billiards with rational caustics,”
Nonlinearity, vol. 31, no. 11. IOP Publishing, pp. 5214–5234, 2018.
ista: Kaloshin V, Zhang K. 2018. Density of convex billiards with rational caustics.
Nonlinearity. 31(11), 5214–5234.
mla: Kaloshin, Vadim, and Ke Zhang. “Density of Convex Billiards with Rational Caustics.”
Nonlinearity, vol. 31, no. 11, IOP Publishing, 2018, pp. 5214–34, doi:10.1088/1361-6544/aadc12.
short: V. Kaloshin, K. Zhang, Nonlinearity 31 (2018) 5214–5234.
date_created: 2020-09-17T10:42:09Z
date_published: 2018-10-15T00:00:00Z
date_updated: 2021-01-12T08:19:10Z
day: '15'
doi: 10.1088/1361-6544/aadc12
extern: '1'
external_id:
arxiv:
- '1706.07968'
intvolume: ' 31'
issue: '11'
keyword:
- Mathematical Physics
- General Physics and Astronomy
- Applied Mathematics
- Statistical and Nonlinear Physics
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://arxiv.org/abs/1706.07968
month: '10'
oa: 1
oa_version: Preprint
page: 5214-5234
publication: Nonlinearity
publication_identifier:
issn:
- 0951-7715
- 1361-6544
publication_status: published
publisher: IOP Publishing
quality_controlled: '1'
status: public
title: Density of convex billiards with rational caustics
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 31
year: '2018'
...
---
_id: '8498'
abstract:
- lang: eng
text: "In the present note we announce a proof of a strong form of Arnold diffusion
for smooth convex Hamiltonian systems. Let ${\\mathbb T}^2$ be a 2-dimensional
torus and B2 be the unit ball around the origin in ${\\mathbb R}^2$ . Fix ρ >
0. Our main result says that for a 'generic' time-periodic perturbation of an
integrable system of two degrees of freedom $H_0(p)+\\varepsilon H_1(\\theta,p,t),\\quad
\\ \\theta\\in {\\mathbb T}^2,\\ p\\in B^2,\\ t\\in {\\mathbb T}={\\mathbb R}/{\\mathbb
Z}$ , with a strictly convex H0, there exists a ρ-dense orbit (θε, pε, t)(t) in
${\\mathbb T}^2 \\times B^2 \\times {\\mathbb T}$ , namely, a ρ-neighborhood of
the orbit contains ${\\mathbb T}^2 \\times B^2 \\times {\\mathbb T}$ .\r\n\r\nOur
proof is a combination of geometric and variational methods. The fundamental elements
of the construction are the usage of crumpled normally hyperbolic invariant cylinders
from [9], flower and simple normally hyperbolic invariant manifolds from [36]
as well as their kissing property at a strong double resonance. This allows us
to build a 'connected' net of three-dimensional normally hyperbolic invariant
manifolds. To construct diffusing orbits along this net we employ a version of
the Mather variational method [41] equipped with weak KAM theory [28], proposed
by Bernard in [7]."
article_processing_charge: No
article_type: original
author:
- first_name: Vadim
full_name: Kaloshin, Vadim
id: FE553552-CDE8-11E9-B324-C0EBE5697425
last_name: Kaloshin
orcid: 0000-0002-6051-2628
- first_name: K
full_name: Zhang, K
last_name: Zhang
citation:
ama: Kaloshin V, Zhang K. Arnold diffusion for smooth convex systems of two and
a half degrees of freedom. Nonlinearity. 2015;28(8):2699-2720. doi:10.1088/0951-7715/28/8/2699
apa: Kaloshin, V., & Zhang, K. (2015). Arnold diffusion for smooth convex systems
of two and a half degrees of freedom. Nonlinearity. IOP Publishing. https://doi.org/10.1088/0951-7715/28/8/2699
chicago: Kaloshin, Vadim, and K Zhang. “Arnold Diffusion for Smooth Convex Systems
of Two and a Half Degrees of Freedom.” Nonlinearity. IOP Publishing, 2015.
https://doi.org/10.1088/0951-7715/28/8/2699.
ieee: V. Kaloshin and K. Zhang, “Arnold diffusion for smooth convex systems of two
and a half degrees of freedom,” Nonlinearity, vol. 28, no. 8. IOP Publishing,
pp. 2699–2720, 2015.
ista: Kaloshin V, Zhang K. 2015. Arnold diffusion for smooth convex systems of two
and a half degrees of freedom. Nonlinearity. 28(8), 2699–2720.
mla: Kaloshin, Vadim, and K. Zhang. “Arnold Diffusion for Smooth Convex Systems
of Two and a Half Degrees of Freedom.” Nonlinearity, vol. 28, no. 8, IOP
Publishing, 2015, pp. 2699–720, doi:10.1088/0951-7715/28/8/2699.
short: V. Kaloshin, K. Zhang, Nonlinearity 28 (2015) 2699–2720.
date_created: 2020-09-18T10:46:43Z
date_published: 2015-06-30T00:00:00Z
date_updated: 2021-01-12T08:19:41Z
day: '30'
doi: 10.1088/0951-7715/28/8/2699
extern: '1'
intvolume: ' 28'
issue: '8'
keyword:
- Mathematical Physics
- General Physics and Astronomy
- Applied Mathematics
- Statistical and Nonlinear Physics
language:
- iso: eng
month: '06'
oa_version: None
page: 2699-2720
publication: Nonlinearity
publication_identifier:
issn:
- 0951-7715
- 1361-6544
publication_status: published
publisher: IOP Publishing
quality_controlled: '1'
status: public
title: Arnold diffusion for smooth convex systems of two and a half degrees of freedom
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 28
year: '2015'
...
---
_id: '8527'
abstract:
- lang: eng
text: We introduce a new potential-theoretic definition of the dimension spectrum of
a probability measure for q > 1 and explain its relation to prior definitions.
We apply this definition to prove that if and is a Borel probability measure
with compact support in , then under almost every linear transformation from to
, the q-dimension of the image of is ; in particular, the q-dimension of is
preserved provided . We also present results on the preservation of information
dimension and pointwise dimension. Finally, for and q > 2 we give examples for
which is not preserved by any linear transformation into . All results for typical
linear transformations are also proved for typical (in the sense of prevalence)
continuously differentiable functions.
article_processing_charge: No
article_type: original
author:
- first_name: Brian R
full_name: Hunt, Brian R
last_name: Hunt
- first_name: Vadim
full_name: Kaloshin, Vadim
id: FE553552-CDE8-11E9-B324-C0EBE5697425
last_name: Kaloshin
orcid: 0000-0002-6051-2628
citation:
ama: Hunt BR, Kaloshin V. How projections affect the dimension spectrum of fractal
measures. Nonlinearity. 1997;10(5):1031-1046. doi:10.1088/0951-7715/10/5/002
apa: Hunt, B. R., & Kaloshin, V. (1997). How projections affect the dimension
spectrum of fractal measures. Nonlinearity. IOP Publishing. https://doi.org/10.1088/0951-7715/10/5/002
chicago: Hunt, Brian R, and Vadim Kaloshin. “How Projections Affect the Dimension
Spectrum of Fractal Measures.” Nonlinearity. IOP Publishing, 1997. https://doi.org/10.1088/0951-7715/10/5/002.
ieee: B. R. Hunt and V. Kaloshin, “How projections affect the dimension spectrum
of fractal measures,” Nonlinearity, vol. 10, no. 5. IOP Publishing, pp.
1031–1046, 1997.
ista: Hunt BR, Kaloshin V. 1997. How projections affect the dimension spectrum of
fractal measures. Nonlinearity. 10(5), 1031–1046.
mla: Hunt, Brian R., and Vadim Kaloshin. “How Projections Affect the Dimension Spectrum
of Fractal Measures.” Nonlinearity, vol. 10, no. 5, IOP Publishing, 1997,
pp. 1031–46, doi:10.1088/0951-7715/10/5/002.
short: B.R. Hunt, V. Kaloshin, Nonlinearity 10 (1997) 1031–1046.
date_created: 2020-09-18T10:50:41Z
date_published: 1997-06-19T00:00:00Z
date_updated: 2021-01-12T08:19:53Z
day: '19'
doi: 10.1088/0951-7715/10/5/002
extern: '1'
intvolume: ' 10'
issue: '5'
keyword:
- Mathematical Physics
- General Physics and Astronomy
- Applied Mathematics
- Statistical and Nonlinear Physics
language:
- iso: eng
month: '06'
oa_version: None
page: 1031-1046
publication: Nonlinearity
publication_identifier:
issn:
- 0951-7715
- 1361-6544
publication_status: published
publisher: IOP Publishing
quality_controlled: '1'
status: public
title: How projections affect the dimension spectrum of fractal measures
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 10
year: '1997'
...