A note on the vanishing viscosity limit in the Yudovich class

TL;DR

This paper investigates the inviscid limit of 2D Navier--Stokes equations with bounded vorticity, providing a refined bound on the difference between Navier--Stokes and Euler solutions as viscosity tends to zero.

Contribution

It offers an improved bound on the convergence rate in the vanishing viscosity limit within the Yudovich class, refining previous results by Chemin.

Findings

Derived a bound of order ( u/|log u|)^{1/2exp(-Ct)} for the difference between Navier--Stokes and Euler velocities.

Extended understanding of the inviscid limit in the Yudovich class with a more precise convergence rate.

Provided mathematical evidence supporting the expected vanishing difference in the inviscid limit.

Abstract

We consider the inviscid limit for the two-dimensional Navier--Stokes equations in the class of integrable and bounded vorticity fields. It is expected that the difference between the Navier--Stokes and Euler velocity fields vanishes in $L^2$ with an order proportional to the square root of the viscosity constant $\nu$. Here, we provide an order $\left(\nu/|\log\nu|\right)^{\frac12\exp(-Ct)}$ bound, which slightly improves upon earlier results by Chemin.