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

Vorticity and stream function formulations for the 2D Navier-Stokes equations in a bounded domain