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Vorticity and stream function formulations for the 2D Navier-Stokes equations in a bounded domain. (English) Zbl 1433.76039

Summary: The main purpose of this work is to provide a Hilbertian functional framework for the analysis of the planar incompressible 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 Y. Maekawa [Adv. Differ. Equ. 18, No. 1–2, 101–146 (2013; Zbl 1261.35111)]. 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.

MSC:

76D05 Navier-Stokes equations for incompressible viscous fluids
76D17 Viscous vortex flows

Citations:

Zbl 1261.35111
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References:

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