On the well-posedness of a class of non-autonomous SPDEs: an operator-theoretical perspective

Rainer Picard, Sascha Trostorff, Marcus Waurick

Research output: Contribution to journalArticle

Abstract

We further elaborate on the solvability of stochastic partial differential equations (SPDEs). We shall discuss non-autonomous partial differential equations with an abstract realization of the stochastic integral on the right-hand side. Our approach allows the treatment of equations with mixed type, where classical solution strategies fail to work. The approach extends prior observations in [Suß, A. & Waurick, M. A Solution Theory for a General Class of SPDEs. Stochastics and Partial Differential Equations: Analysis and Computations, 2017, 5, 278-318], where the respective results were obtained for linear autonomous equations and (multiplicative) white noise.
LanguageEnglish
Article numbere201800014
Number of pages16
JournalGAMM-Mitteilungen / GAMM-Reports
Early online date12 Oct 2018
DOIs
Publication statusE-pub ahead of print - 12 Oct 2018

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Stochastic Partial Differential Equations
Well-posedness
partial differential equations
Partial differential equations
Mathematical operators
Partial differential equation
operators
Nonautonomous Differential Equations
Stochastic Integral
Operator
Classical Solution
White noise
Solvability
Multiplicative
white noise
Class
Observation
Strategy

Keywords

  • stochastic partial differential equations
  • evolutionary equations
  • stochastic equations of mathematical physics
  • weak solutions
  • non-autonomous equations
  • nonlinear equations
  • differential inclusions

Cite this

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abstract = "We further elaborate on the solvability of stochastic partial differential equations (SPDEs). We shall discuss non-autonomous partial differential equations with an abstract realization of the stochastic integral on the right-hand side. Our approach allows the treatment of equations with mixed type, where classical solution strategies fail to work. The approach extends prior observations in [Su{\ss}, A. & Waurick, M. A Solution Theory for a General Class of SPDEs. Stochastics and Partial Differential Equations: Analysis and Computations, 2017, 5, 278-318], where the respective results were obtained for linear autonomous equations and (multiplicative) white noise.",
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AU - Trostorff, Sascha

AU - Waurick, Marcus

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AB - We further elaborate on the solvability of stochastic partial differential equations (SPDEs). We shall discuss non-autonomous partial differential equations with an abstract realization of the stochastic integral on the right-hand side. Our approach allows the treatment of equations with mixed type, where classical solution strategies fail to work. The approach extends prior observations in [Suß, A. & Waurick, M. A Solution Theory for a General Class of SPDEs. Stochastics and Partial Differential Equations: Analysis and Computations, 2017, 5, 278-318], where the respective results were obtained for linear autonomous equations and (multiplicative) white noise.

KW - stochastic partial differential equations

KW - evolutionary equations

KW - stochastic equations of mathematical physics

KW - weak solutions

KW - non-autonomous equations

KW - nonlinear equations

KW - differential inclusions

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