2D Navier-Stokes with Navier Slip: Strong Vorticity Convergence and Strong Solutions for Unbounded Vorticity
Josef Demmel, Emil Wiedemann
TL;DR
This work studies the vanishing viscosity limit for the 2D incompressible Navier–Stokes equations with Navier slip on a smooth bounded domain, for initial vorticity in $L^{p}$ with $p>2$. The authors blend an interior framework originally developed for no-slip boundaries with Agmon–Douglas–Nirenberg elliptic theory for the Laplacian under Navier boundary conditions, employing a compatible-data approximation to upgrade local results to global strong convergence of the vorticity and to obtain strong solutions for the viscous problem. They establish strong convergence of the vorticity $\omega^{\nu}$ to a Euler vorticity $\omega$ in $C([0,T];L^{q}(\Omega))$ for all $q\in[1,p)$, and prove that the velocity $u^{\nu}$ becomes a strong solution satisfying the Navier boundary condition for all positive times. This work removes boundary-layer obstructions associated with no-slip boundaries and provides a robust framework for strong regularity and inviscid limits in 2D with Navier slip.
Abstract
We analyze the two-dimensional incompressible Navier-Stokes equations on a smooth, bounded domain with Navier boundary conditions. Starting from an initial vorticity in $L^p$ with $p>2$, we show strong convergence of the vorticity in the vanishing viscosity limit. We utilize a purely interior framework from Seis, Wiedemann, and Woźnicki, originally derived for no-slip, and upgrade local to global convergence. Under the same assumptions, we also show that the velocity is in fact a strong solution and satisfies the Navier slip conditions for any positive time. The key idea is to study the Laplacian subject to Navier boundary conditions and prove that this boundary-value problem is elliptic in the sense of Agmon-Douglis-Nirenberg.