Moderate deviation principles for stochastic 2D hydrodynamics type systems with multiplicative Lévy noise

In this paper, we establish a moderate deviation principle for an abstract nonlinear equation forced by random noise of Lévy type. This type of equation covers many hydrodynamical models, including stochastic 2D Navier-Stokes equations, stochastic 2D MHD equations, the stochastic 2D magnetic Bérnard problem, and also several stochastic shell models of turbulence. This paper gets rid of the compact embedding assumption on the associated Gelfand triple. The weak convergence method plays an important role.