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

TL;DR

This paper proves local and global existence and uniqueness of solutions for nonlinear stochastic evolution equations under generalized coercivity conditions, simplifying known results and establishing new ones for complex PDE models.

Contribution

It introduces generalized coercivity conditions for well-posedness of nonlinear SPDEs, extending existing theories and applying them to various complex models.

Findings

Established well-posedness for stochastic 3D Navier-Stokes equations.

Provided simpler proofs for known PDE cases.

Derived new results for stochastic surface growth and power law fluids.

Abstract

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.