The critical variational setting for stochastic evolution equations

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.

• Publication year

  2024

• Venue

  Probability theory and related fields

• Publication date

  2024-02-02

• Fields of study

  Not labeled

• Identifiers
  DOI 10.1007/s00440-023-01249-x
• External record

  Open on Semantic Scholar

• Source metadata

  Semantic Scholar

• No claims are published for this paper.

• No concepts are published for this paper.

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