Stochastic hydrodynamic-type evolution equations driven by Lévy noise in 3D unbounded domains—Abstract framework and applications

The existence of martingale solutions of the hydrodynamic-type equations in 3D possibly unbounded domains is proved. The construction of the solution is based on the Faedo–Galerkin approximation. To overcome the difficulty related to the lack of the compactness of Sobolev embeddings in the case of unbounded domain we use certain Fréchet space. Besides, we use compactness and tightness criteria in some nonmetrizable spaces and a version of the Skorohod theorem in non-metric spaces. The general framework is applied to the stochastic Navier–Stokes, magneto-hydrodynamic (MHD) and the Boussinesq equations.