# Vanishing viscosity limit for global attractors for the damped Navier--Stokes system with stress free boundary conditions

## TL;DR

This paper proves that as viscosity vanishes, the global attractors of the damped Navier--Stokes system with stress-free boundary conditions converge to the attractor of the damped Euler system in a bounded domain, establishing a connection between viscous and inviscid models.

## Contribution

It demonstrates the convergence of global attractors for the damped Navier--Stokes system to the Euler system's attractor in the vanishing viscosity limit under stress-free boundary conditions.

## Key findings

- Global attractor exists for the damped Euler system in H^1.
- Attractors of Navier--Stokes converge to Euler attractor as viscosity vanishes.
- Convergence is in the non-symmetric Hausdorff distance in H^1.

## Abstract

We consider the damped and driven Navier--Stokes system with stress free boundary conditions and the damped Euler system in a bounded domain $\Omega\subset\mathbf{R}^2$. We show that the damped Euler system has a (strong) global attractor in~$H^1(\Omega)$. We also show that in the vanishing viscosity limit the global attractors of the Navier--Stokes system converge in the non-symmetric Hausdorff distance in $H^1(\Omega)$ to the the strong global attractor of the limiting damped Euler system (whose solutions are not necessarily unique).