---
_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. <i>Nonlinearity</i>.
    2022;35(8):4100-4210. doi:<a href="https://doi.org/10.1088/1361-6544/abd613">10.1088/1361-6544/abd613</a>
  apa: Agresti, A., &#38; Veraar, M. (2022). Nonlinear parabolic stochastic evolution
    equations in critical spaces Part I. Stochastic maximal regularity and local existence.
    <i>Nonlinearity</i>. IOP Publishing. <a href="https://doi.org/10.1088/1361-6544/abd613">https://doi.org/10.1088/1361-6544/abd613</a>
  chicago: Agresti, Antonio, and Mark Veraar. “Nonlinear Parabolic Stochastic Evolution
    Equations in Critical Spaces Part I. Stochastic Maximal Regularity and Local Existence.”
    <i>Nonlinearity</i>. IOP Publishing, 2022. <a href="https://doi.org/10.1088/1361-6544/abd613">https://doi.org/10.1088/1361-6544/abd613</a>.
  ieee: A. Agresti and M. Veraar, “Nonlinear parabolic stochastic evolution equations
    in critical spaces Part I. Stochastic maximal regularity and local existence,”
    <i>Nonlinearity</i>, 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.”
    <i>Nonlinearity</i>, vol. 35, no. 8, IOP Publishing, 2022, pp. 4100–210, doi:<a
    href="https://doi.org/10.1088/1361-6544/abd613">10.1088/1361-6544/abd613</a>.
  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
license: https://creativecommons.org/licenses/by/3.0/
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'
...