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Abstract

We study the existence and uniqueness of strong solutions, in the sense of PDEs and probability, for a broad class of nonlinear stochastic partial differential equations (SPDEs) on a bounded domain $\mathscr{O}\subset \mathbb{R}^d$ ($d\leq 3$), perturbed by spatially correlated multiplicative noise. Our framework applies to many physically relevant systems, including stochastic convective Brinkman--Forchheimer equations, stochastic magnetohydrodynamics (MHD), stochastic B\'enard convection in porous media, stochastic convective dynamo models, and stochastic magneto-micropolar fluids, among others. The analysis relies on Galerkin approximations and compactness arguments. Up to a suitable stopping time, we derive strong moment bounds and verify a Cauchy property for the approximate solutions, in the absence of any inherent cancellation structure. By applying a Gronwall-type lemma for stochastic processes, we establish the existence and uniqueness of maximal strong pathwise solutions, which are global in two spatial dimensions. These results provide a unified treatment of a wide class of nonlinear stochastic thermo-magneto-fluid models in the literature.