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arXiv · 2611.05212

2D Navier-Stokes with viscosity-dependent Navier slip: strong convergence for unbounded initial velocity

Josef Demmel
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Abstract

For the two-dimensional incompressible Navier-Stokes equations with Navier friction on a smooth, bounded, simply connected domain, we establish strong vorticity convergence along a subsequence in the vanishing viscosity limit for initial vorticity in $L^p$ with $p>1$. The friction coefficients may grow as viscosity vanishes under an explicit growth restriction depending on $p$, which includes regimes of vanishing slip length. The main challenge, even for viscosity-independent friction, is to control the vorticity generated at the boundary. We split the vorticity into contributions from the initial data and the boundary. By combining renormalization and estimates for the normal velocity near the boundary, we show that the boundary contribution vanishes in the inviscid limit. As a consequence, the limiting Euler solutions conserve kinetic energy.

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Josef Demmel. 2026-11-03. 2D Navier-Stokes with viscosity-dependent Navier slip: strong convergence for unbounded initial velocity. https://arxiv.org/abs/2611.05212

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