Global pathwise weak solution for two-dimensional stochastic Keller-Segel(-Navier)-Stokes system with logistic source

This paper addresses the global solvability of a stochastic Keller-Segel(-Navier)-Stokes [SKS(N)S, for short] system with logistic source in a smooth, bounded domain O⊂R2. The main results consist of two parts. First, we prove the existence of a unique global pathwise weak solution for the SKSS system, where the chemical concentration equation is affected by small conservative noise, and the Stokes (slow fluid) equations are perturbed by multiplicative Gaussian noise. Second, we show that the SKSNS system also admits a unique global pathwise weak solution when only the Navier-Stokes equations are influenced by multiplicative Gaussian noise. To establish these results, we use the Galerkin approximation scheme combined with the stochastic compactness method.