Local martingale and pathwise solutions for an abstract fluids model

Abstract

We establish the existence and uniqueness of both local martingale and local pathwise solutions of an abstract nonlinear stochastic evolution system. The primary application of this abstract framework is to infer the local existence of strong, pathwise solutions to the 3D primitive equations of the oceans and atmosphere forced by a nonlinear multiplicative white noise. Instead of developing our results, specifically for the 3D primitive equations we choose to develop them in a slightly abstract framework which covers many related forms of these equations (atmosphere, oceans, coupled atmosphere–ocean, on the sphere, on the β -plane approximation etc. and the incompressible Navier–Stokes equations). In applications, all the details are given for the β -plane approximation of the equations of the ocean.