Local and global well-posedness of SPDE with generalized coercivity conditions

In this paper we establish the 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 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.