2D Navier-Stokes with Navier Slip: Strong Vorticity Convergence and Strong Solutions for Unbounded Vorticity

TL;DR

This paper proves strong vorticity convergence and existence of strong solutions for 2D Navier-Stokes equations with Navier slip boundary conditions, starting from unbounded initial vorticity, using an interior analysis approach.

Contribution

It extends the analysis of Navier-Stokes equations with Navier slip to unbounded vorticity and establishes strong convergence and solution regularity in this setting.

Findings

Strong vorticity convergence in vanishing viscosity limit

Velocity becomes a strong solution satisfying Navier slip conditions

Elliptic regularity results for Laplacian with Navier boundary conditions

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 Wo\'{z}nicki, 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.