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This paper considers the Keller-Segel model coupled to stochasticNavier-Stokes equations (KS-SNS, for short), which describes the dynamics ofoxygen and bacteria densities evolving within a stochastically forcedincompressible viscous flow. Our main results establish the existence anduniqueness results of global solutions (strong in probabilistic sense and weakin PDE's sense) to the KS-SNS system in $\mathbb{R}^2$. A novel approximateKS-SNS system with proper regularization and cut-off operators in$H^s(\mathbb{R}^2)$ is introduced, and the existence of approximate solution isproved by some a priori uniform bounds and a careful analysis on theapproximation scheme. Under different assumptions, two types of entropy-energyfunctional inequality that seems to be new in their forms for the regularizedKS-SNS system are derived, which together with the Prohorov's theorem andJakubowski-Skorokhod theorem enable us to prove that the sequence ofapproximate solutions converges to a global martingale weak solution to theKS-SNS system. Moreover, when $\chi(\cdot)\equiv \textrm{const.}>0$, we provethat the solution is pathwise unique, and hence by the Yamada-Wantanabe theoremthat the KS-SNS system admits a unique global pathwise weak solution.

On the Keller-Segel models interacting with a stochastically forced incompressible viscous flow in $\mathbb{R}^2$