Regularity of the global attractor for the 2D incompressible Navier-Stokes equations on channel-like domains

TL;DR

This paper proves that the global attractor for 2D incompressible Navier-Stokes equations on channel-like domains is regular and compact in higher Sobolev spaces, under minimal boundary regularity assumptions.

Contribution

It establishes the regularity and compactness of the global attractor in Sobolev spaces for flows on general channel-like domains without boundary regularity assumptions.

Findings

Global attractor is compact in V space.

Solutions converge to the attractor in V norm.

Attractor is compact in higher regularity spaces under forcing conditions.

Abstract

The regularity of the global attractor of the incompressible Navier-Stokes equations for flows on two-dimensional domains is considered. It is assumed that domain $\Omega$ is a channel-like domain, i.e an arbitrary bounded or unbounded domain, at first without any regularity assumption on its boundary, with the only assumption that the Poincar\'e inequality holds on it. The phase space $H$ for the system is the usual closure in the $L^2(\Omega)^2$ norm of the space of smooth divergent-free vector-fields with compact support in $\Omega.$ The corresponding space obtained as the closure with respect to the $H^1(\Omega)^2$ norm is denoted by $V.$ The forcing term is assumed to belong to dual space $V'.$ It is known in this case that the global attractor exists in the phase space $H.$ It is shown in this work that the global attractor is also a compact set in $V$, and that due to the regularization effect of the equations, the solutions converge to the attractor in the norm of $V$, uniformly for initial conditions bounded in $H$. Moreover, it is shown that if the forcing term belongs to $D(A^{-s})$, for some $0<s\leq 1/2$, where $A$ is the Stokes operator, then the global attractor is compact in $D(A^{-s+1})$. If the forcing term is in $H,$ corresponding to the limit case $s=0,$ then it is further assumed that the domain is either a uniformly $\Ccal^{1,1}$ smooth domain or a bounded Lipschitz domain in order to obtain that the global attractor is compact in $D(A).$