  Nonlinear Fokker-Planck equations endowed with curl drift forces are
investigated. The conditions under which these evolution equations admit
stationary solutions, which are $q$-exponentials of an appropriate potential
function, are determined. It is proved that when these stationary solutions
exist, the nonlinear Fokker-Planck equations satisfy an $H$-theorem in terms of
a free-energy like quantity involving the $S_q$ entropy. A particular two
dimensional model admitting analytical, time-dependent, $q$-Gaussian solutions
is discussed in detail. This model describes a system of particles with
short-range interactions, performing overdamped motion under drag effects, due
to a rotating resisting medium. It is related to models that have been recently
applied to the study of type-II superconductors. The relevance of the present
developments to the study of complex systems in physics, astronomy, and
biology, is discussed.
