  We consider a nonlocal nonlinear model with fractional diffusion motivated by
studies of electroconvection phenomena in incompressible viscous fluids. We
address the global well-posedness, global regularity and long time dynamics of
the model in bounded smooth domains with Dirichlet boundary conditions. We
prove the existence and uniqueness of exponentially decaying in time solutions
for $H^1$ initial data regardless of the fractional dissipative regularity. In
the presence of time independent body forces in the fluid, we prove the
existence of a compact finite dimensional global attractor. In the case of
periodic boundary conditions, we prove that the unique smooth solution is
globally analytic in time, and belongs to a Gevrey class of functions that
depends on the dissipative regularity of the model.
