Upper semicontinuous attractors for 3D hyperviscous flow

TL;DR

This paper investigates how the global attractors of regularized 3D Navier-Stokes equations behave as the added fourth-order viscosity diminishes, with implications for understanding fluid flow stability under perturbations.

Contribution

It establishes the upper semicontinuity of attractors for the regularized equations as the artificial dissipation parameter approaches zero.

Findings

Proves upper semicontinuity of attractors in the regularized 3D Navier-Stokes equations.

Analyzes the effect of varying forcing functions on the attractors.

Provides insights into the stability of fluid flows with added higher-order viscosity.

Abstract

We regularized the 3D Navier-Stokes equations by adding a fourth-order viscosity term. We study the upper semicontinuity, of the global attractors of the Leray-Hopf weak solutions of the regularized 3D Navier-Stokes equations, as the artificial dissipation goes to 0. We also consider applications of obtained results to the regularized problem by allowing the family of forcing functions to vary with epsilon, for epsilon >0.