Pullback dynamics of 2D incompressible non-autonomous Navier-Stokes equation on Lipschitz-like domain

TL;DR

This paper studies the long-term behavior of solutions to the 2D non-autonomous Navier-Stokes equations on Lipschitz-like domains, establishing the existence, properties, and regularity of pullback attractors under various conditions.

Contribution

It proves the existence of minimal pullback attractors for non-autonomous Navier-Stokes equations on Lipschitz-like domains and analyzes their dimension, semi-continuity, and regularity.

Findings

Existence of minimal pullback attractors for the system.

Finite fractal dimension estimates of the attractors.

Upper semi-continuity of attractors under perturbations.

Abstract

This paper concerns the tempered pullback dynamics of 2D incompressible non-autonomous Navier-Stokes equation with non-homogeneous boundary condition on Lipschitz-like domain. With the presence of a time-dependent external force f(t) which only needs to be pullback translation bounded, we establish the existence of a minimal pullback attractor with respect to a universe of tempered sets for the corresponding non-autonomous dynamical system. We then give estimate on the finite fractal dimension of the attractor based on trace formula. Under the additional assumption that the external force is the sum of a stationary force and a non-autonomous perturbation, we also prove the upper semi-continuity of the attractors as the non-autonomous perturbation vanishes. Lastly, we also investigate the regularity of these attractors when smoother initial data is given. Our results are new even in the case of smooth domains.