The critical variational setting for stochastic evolution equations

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. Moreover, 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 equations, tamed Navier-Stokes equations, and Allen-Cahn equation.

The stochastic primitive equations with non-isothermal turbulent pressure

“…Proof of Theorem 3.7. To prove Theorem 3.7 we argue as in [AV22a]. As in the proof of Theorem 3.7 readily follows from the following result.…”

Reaction-Diffusion Equations with Transport Noise and Critical Superlinear Diffusion: Local Well-Posedness and Positivity

“…The condition (1.3) can be seen as an L ζ -version of the usual coercivity condition for stochastic evolution equations in the variational setting, see e.g. [AV22a], [LR15, Chapter 4], and [RSZ22]. The restriction ζ ≥ d 2 (h − 1) is related to the aforementioned criticality of L d 2 (h−1) for (1.1) (see [AV22d, Subsection 1.4]).…”