Well-posedness and large deviations for 2D stochastic constrained Navier-Stokes equations driven by Lévy noise in the Marcus canonical form

We consider stochastic two-dimensional constrained Navier-Stokes equations driven by Levy noise in the Marcus canonical form. The aim of this work is two-fold. At first, we prove the existence of a martingale solution based on the construction relying on classical Faedo-Galerkin approximations, compactness method and the Jakubowski's version of Skorokhod representation theorem for non-metric spaces. We further prove that the martingale solution is pathwise unique and deduces the existence of a strong solution. In the second part of the paper, we establish a Wentzell-Freidlin type large deviations principle for the small noise asymptotic of solutions using weak convergence method.