On the inviscid limit for 2D incompressible flow with Navier friction condition

TL;DR

This paper extends previous work on the inviscid limit of 2D incompressible Navier-Stokes flows with Navier friction boundary conditions, accommodating initial vorticities with higher integrability ($p>2$).

Contribution

It adapts and simplifies existing proofs to include initial vorticities in $L^p$ spaces for $p>2$, broadening the class of flows analyzed.

Findings

Inclusion of $p$-th power integrable initial vorticities in the inviscid limit analysis.

Simplified proof techniques for the inviscid limit with Navier friction boundary conditions.

Confirmation that the inviscid limit satisfies the Euler equations under broader initial conditions.

Abstract

In [1], T. Clopeau, A. Mikeli\'c, and R. Robert studied the inviscid limit of the 2D incompressible Navier-Stokes equations in a bounded domain subject to Navier friction-type boundary conditions. They proved that the inviscid limit satisfies the incompressible Euler equations and their result ultimately includes flows generated by bounded initial vorticities. Our purpose in this article is to adapt and, to some extent, simplify their argument in order to include $p$-th power integrable initial vorticities, with $p>2$. [1] Clopeau, T., Mikeli\'c, A., Robert, R., {\it On the vanishing viscosity limit for the 2D incompressible Navier-Stokes equations with the friction type boundary conditions}, Nonlinearity {\bf 11} (1998) 1625--1636.