The critical variational setting for stochastic evolution equations

Antonio Agresti, Mark Veraar*

*Corresponding author for this work

Research output: Contribution to journalArticleScientificpeer-review

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Abstract

In this paper we introduce the critical variational setting for parabolic stochastic evolution equations of quasi- or semi-linear type. Our results improve many of the abstract results in the classical variational setting. In particular, we are able to replace the usual weak or local monotonicity condition by a more flexible local Lipschitz condition. Moreover, the usual growth conditions on the multiplicative noise are weakened considerably. Our new setting provides general conditions under which local and global existence and uniqueness hold. In addition, we prove continuous dependence on the initial data. We show that many classical SPDEs, which could not be covered by the classical variational setting, do fit in the critical variational setting. In particular, this is the case for the Cahn–Hilliard equation, tamed Navier–Stokes equations, and Allen–Cahn equation.

Original languageEnglish
Pages (from-to)957-1015
Number of pages59
JournalProbability Theory and Related Fields
Volume188
Issue number3-4
DOIs
Publication statusPublished - 2024

Keywords

  • Allen–Cahn equation
  • Cahn–Hilliard equation
  • Coercivity
  • Critical nonlinearities
  • Generalized Burgers equation
  • Quasi- and semi-linear
  • Stochastic evolution equations
  • Stochastic partial differential equations
  • Swift–Hohenberg equation
  • Tamed Navier–Stokes
  • Variational methods

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