Uniform global attractors for the nonautonomous 3D Navier-Stokes equations

TL;DR

This paper establishes the existence and structure of uniform global attractors for the nonautonomous 3D Navier-Stokes equations under certain conditions, providing a general framework applicable to other dissipative PDEs.

Contribution

It introduces a novel method for analyzing nonautonomous 3D Navier-Stokes equations without requiring solution uniqueness, extending attractor theory to broader classes of PDEs.

Findings

Existence of weak uniform global attractors for nonautonomous 3D NSE.

Conditions under which the attractor is strong and solutions converge strongly.

Framework applicable to other nonautonomous dissipative PDEs without uniqueness.

Abstract

We obtain the existence and the structure of the weak uniform (with respect to the initial time) global attractor and construct a trajectory attractor for the 3D Navier-Stokes equations (NSE) with a fixed time-dependent force satisfying a translation boundedness condition. Moreover, we show that if the force is normal and every complete bounded solution is strongly continuous, then the uniform global attractor is strong, strongly compact, and solutions converge strongly toward the trajectory attractor. Our method is based on taking a closure of the autonomous evolutionary system without uniqueness, whose trajectories are solutions to the nonautonomous 3D NSE. The established framework is general and can also be applied to other nonautonomous dissipative partial differential equations for which the uniqueness of solutions might not hold. It is not known whether previous frameworks can also be applied in such cases as we indicate in open problems related to the question of uniqueness of the Leray-Hopf weak solutions.