  In this paper we establish local and global existence and uniqueness of
solutions for general nonlinear evolution equations with coefficients
satisfying some local monotonicity and generalized coercivity conditions. An
analogous result is obtained for stochastic evolution equations in Hilbert
space with general additive noise. As applications, the main results are
applied to obtain simpler proofs in known cases as the stochastic 3D
Navier-Stokes equation, the tamed 3D Navier-Stokes equation and the
Cahn-Hilliard equation, but also to get new results for stochastic surface
growth PDE and stochastic power law fluids.
