Nonlinear parabolic stochastic evolution equations in critical spaces Part I. Stochastic maximal regularity and local existence

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.

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Author
Agresti, AntonioISTA ; Veraar, Mark
Department
Abstract
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.
Publishing Year
Date Published
2022-08-04
Journal Title
Nonlinearity
Acknowledgement
The second author is supported by the VIDI subsidy 639.032.427 of the Netherlands Organisation for Scientific Research (NWO).
Volume
35
Issue
8
Page
4100-4210
ISSN
eISSN
IST-REx-ID

Cite this

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
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
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.
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.
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.
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.
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