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

TL;DR

This paper develops a Hilbertian framework for analyzing 2D Navier-Stokes equations using vorticity and stream function formulations, establishing equivalence with classical methods and deriving new existence, uniqueness, and decay results.

Contribution

It introduces a nonprimitive variable approach for 2D Navier-Stokes analysis, simplifying the vorticity dynamics and proving new long-term decay properties.

Findings

Established equivalence between primitive and nonprimitive variable formulations.

Derived explicit Bernoulli-like formulas for pressure recovery.

Proved exponential decay of vorticity norm (palinstrophy) over time.

Abstract

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.