The critical variational setting for stochastic evolution equations

TL;DR

This paper introduces a new critical variational framework for stochastic evolution equations that broadens the applicability of existing theories, allowing for weaker conditions and covering more complex SPDEs.

Contribution

It develops a more flexible variational setting for SPDEs, replacing traditional monotonicity with local Lipschitz conditions and weakening growth assumptions on noise.

Findings

Established conditions for local and global existence and uniqueness.

Proved continuous dependence on initial data.

Included classical SPDEs like Cahn-Hilliard and Navier-Stokes in the new framework.

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.