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We present two criteria to conclude that a stochastic partial differentialequation (SPDE) posseses a unique maximal strong solution. This paper providesthe full details of the abstract well-posedness results first given inarXiv:2202.09242v2, and partners a paper which rigorously addressesapplications to the 3D SALT (Stochastic Advection by Lie Transport)Navier-Stokes Equation in velocity and vorticity form, on the torus and thebounded domain respectively. Each criterion has its corresponding set ofassumptions and can be applied to viscous fluid equations with additive,multiplicative or a general transport type noise.

Existence and Uniqueness of Maximal Solutions to SPDEs with Applications to Viscous Fluid Equations