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

Ricardo M. S. Rosa

TL;DR

This work analyzes the regularity of the global attractor for the 2D incompressible Navier–Stokes equations on channel-like domains with minimal boundary assumptions, assuming only the Poincaré inequality. It shows the attractor is compact in the energy space $H$ and, due to the regularization effect of the equations, compact in the stronger space $V$; further regularity of the forcing $f$ yields compactness in $D(A^{-s+1})$ for $0< s\leq \tfrac{1}{2}$, and, under domain regularity, compactness in $D(A)$ when $f\in H$. The proofs combine semigroup regularization techniques with energy-type methods to establish asymptotic compactness and continuity properties, including higher-regularity results under two regimes for the forcing and domain. These results are optimal with respect to the available assumptions and extend classical NSE attractor theory to non-smooth, unbounded channel-like domains. The findings have implications for understanding long-term flow behavior and for precision in numerical approximations in these general geometries.

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 $Ω$ 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 Poincaré inequality holds on it. The phase space $H$ for the system is the usual closure in the $L^2(Ω)^2$ norm of the space of smooth divergent-free vector-fields with compact support in $Ω.$ The corresponding space obtained as the closure with respect to the $H^1(Ω)^2$ 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 $D(A).$

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

TL;DR

This work analyzes the regularity of the global attractor for the 2D incompressible Navier–Stokes equations on channel-like domains with minimal boundary assumptions, assuming only the Poincaré inequality. It shows the attractor is compact in the energy space and, due to the regularization effect of the equations, compact in the stronger space ; further regularity of the forcing yields compactness in for , and, under domain regularity, compactness in when . The proofs combine semigroup regularization techniques with energy-type methods to establish asymptotic compactness and continuity properties, including higher-regularity results under two regimes for the forcing and domain. These results are optimal with respect to the available assumptions and extend classical NSE attractor theory to non-smooth, unbounded channel-like domains. The findings have implications for understanding long-term flow behavior and for precision in numerical approximations in these general geometries.

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 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 Poincaré inequality holds on it. The phase space for the system is the usual closure in the norm of the space of smooth divergent-free vector-fields with compact support in The corresponding space obtained as the closure with respect to the norm is denoted by The forcing term is assumed to belong to dual space It is known in this case that the global attractor exists in the phase space It is shown in this work that the global attractor is also a compact set in , and that due to the regularization effect of the equations, the solutions converge to the attractor in the norm of , uniformly for initial conditions bounded in . Moreover, it is shown that if the forcing term belongs to , for some , where is the Stokes operator, then the global attractor is compact in . If the forcing term is in corresponding to the limit case then it is further assumed that the domain is either a uniformly smooth domain or a bounded Lipschitz domain in order to obtain that the global attractor is compact in

Paper Structure

This paper contains 13 sections, 5 theorems, 98 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. General settings
  4. More regular domains
  5. Descriptions of the main results
  6. Proofs using the regularization of the semigroup
  7. Regularity of the semigroup
  8. Compactness of the global attractor in V - proof of Theorem \ref{['Vcompactnessthm']}
  9. Compactness in H of the time-derivative of the solutions - proof of Proposition \ref{['timecompactnessprop']}
  10. Higher-order regularity - proofs of Theorems \ref{['higherregularitythm']} and \ref{['DAcompactnessthm']} and of Proposition \ref{['DAcompactnessprop']}
  11. Proofs using energy-type equations
  12. Compactness of the global attractor in V - proof of Theorem \ref{['Vcompactnessthm']}) via the energy method
  13. Compactness in H of the time-derivative of the solutions - proof of Proposition \ref{['timecompactnessprop']}) via the energy method

Key Result

Theorem 3.1

Let $\Omega\subset \mathbb{R}^2$ be an open set satisfying the Poincaré inequality prepoincare. Assume $\nu>0$ and $f\in V'$. Then, the global attractor ${\mathcal{A}}$ in $H$ of the semigroup $\{S(t)\}_{t\ge0}$ associated with the evolution equation weaknse is compact in $V$.

Theorems & Definitions (5)

  • Theorem 3.1
  • Proposition 3.1
  • Theorem 3.2
  • Proposition 3.2
  • Theorem 3.3