  The main purpose of this work is to provide a Hilbertian functional framework
for the analysis of the planar Navier-Stokes (NS) equations either in vorticity
or in stream function formulation. The fluid is assumed to occupy a bounded
possibly multiply connected domain. The velocity field satisfies either
homogeneous (no-slip boundary conditions) or prescribed Dirichlet boundary
conditions. We prove that the analysis of the 2D Navier-Stokes equations can be
carried out in terms of the so-called nonprimitive variables only (vorticity
field and stream function) without resorting to the classical NS theory (stated
in primitive variables, i.e. velocity and pressure fields). Both approaches (in
primitive and nonprimitive variables) are shown to be equivalent for weak
(Leray) and strong (Kato) solutions. Explicit Bernoulli-like formulas are
derived and allow recovering the pressure field from the vorticity fields or
the stream function. In the last section, the functional framework described
earlier leads to a simplified rephrasing of the vorticity dynamics, as
introduced by Maekawa in [52]. At this level of regularity, the vorticity
equation splits into a coupling between a parabolic and an elliptic equation
corresponding respectively to the non-harmonic and harmonic parts of the
vorticity equation. By exploiting this structure it is possible to prove new
existence and uniqueness results, as well as the exponential decay of the
palinstrophy (that is, loosely speaking, the $H^1$ norm of the vorticity) for
large time, an estimate which was not known so far.
