A selection principle for 2D steady Euler flows via the vanishing viscosity limit

TL;DR

The paper addresses selecting physically realizable 2D steady Euler flows by examining the vanishing viscosity limit of steady Navier–Stokes solutions under slip boundary conditions. It develops a stream-function framework combined with Morse–Sard and a total-curvature decomposition to constrain possible limits, proving a rigidity theorem that non-shear flows in a periodic strip must contain a contractible closed streamline. Consequently, in a bounded connected domain the vanishing viscosity limit exists iff the vorticity is constant, while in a periodic strip the limits must be of the form $\mathbf{v}=(c_0+c_1x_2+c_2x_2^2,0)$, with the constant-vorticity case reducing to shear flows. These results generalize the Prandtl–Batchelor paradigm by removing nested-streamline assumptions and provide a complete classification of vanishing viscosity limits in the two key planar geometries, offering a rigorous selection mechanism for physically realizable steady Euler flows.

Abstract

The 2D Euler system, governing inviscid incompressible flow, admits infinitely many steady solutions. To identify which are physically realizable, we investigate the vanishing viscosity limit of the steady Navier-Stokes system. Here, a steady Euler flow with a slip boundary condition is called a vanishing viscosity limit if it is the $L^\infty_{\textup{loc}}\cap H^1_{\textup{loc}}$ limit of steady Navier-Stokes solutions (it is known that strong boundary layers can persist). We then completely classify the vanishing viscosity limit in specific planar domains. In a bounded connected domain, we show that the only vanishing viscosity limit is the constant-vorticity flow. This result does not require the approximating Navier-Stokes solutions to have nested closed streamlines, an essential assumption in the century-old Prandtl-Batchelor theorem. In a strip where the Navier-Stokes velocity (but not the pressure) is periodic in the strip direction, we show that the only vanishing viscosity limits are constant flow, Couette flow, and Poiseuille flow. The proof of the latter result relies on the former and on a powerful rigidity theorem, which asserts that any non-shear steady Euler flow in a periodic strip must have contractible closed streamlines.

Figures (1)

• Figure 1: Illustration for the proof of Proposition \ref{['prop_B']}, Step 4.

Theorems & Definitions (29)

• Definition 1.1
• Theorem 1.1
• Remark 1.1
• Remark 1.2
• Remark 1.3
• Remark 1.4
• Theorem 1.2
• Remark 1.5
• Remark 1.6
• Lemma 2.1