  We establish the existence and uniqueness of solutions to an abstract
nonlinear equation driven by a multiplicative noise of L\'evy type, which
covers many hydrodynamical models including 2D Navier-Stokes equations, 2D MHD
equations, the 2D Magnetic Bernard problem, and several Shell models of
turbulence. In the existing literature on this topic, besides the classical
Lipschitz and one sided linear growth conditions, other assumptions, which
might be untypical, are also required on the coefficients of the stochastic
perturbations. This paper is to get rid of these untypical assumptions. Our
assumption on the coefficients of stochastic perturbations is new even for the
Wiener cases, and in some sense, is shown to be quite sharp. A new cutting-off
argument and energy estimation procedure play an important role in establishing
the existence and uniqueness under this assumption.
