  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.

Probability;Analysis of PDEs;Functional Analysis