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Existence and invariant measure of pullback attractors for 3D Navier-Stokes-Voigt equations with delay

Yuming Qin, Huite Jiang

Abstract

In this paper, we study the long-time dynamics of 3D non-autonomous Navier-Stokes-Voigt(NSV) equations with delay. Inspired by [36], we use the contractive function method to prove the pullback D-asymptotical compactness and existence of the pullback attractors. Furthermore, we verify the regularity of pullback attractors by the method in [14, 43, 47] and there exists a unique family of Borel invariant probability measures which is supported by the pullback attractors.

Existence and invariant measure of pullback attractors for 3D Navier-Stokes-Voigt equations with delay

Abstract

In this paper, we study the long-time dynamics of 3D non-autonomous Navier-Stokes-Voigt(NSV) equations with delay. Inspired by [36], we use the contractive function method to prove the pullback D-asymptotical compactness and existence of the pullback attractors. Furthermore, we verify the regularity of pullback attractors by the method in [14, 43, 47] and there exists a unique family of Borel invariant probability measures which is supported by the pullback attractors.
Paper Structure (5 sections, 14 theorems, 94 equations)

This paper contains 5 sections, 14 theorems, 94 equations.

Table of Contents

  1. Introduction
  2. Global Well-posedness of solutions
  3. Existence of pullback attractors in $E_V^2$ norm
  4. Regularity of pullback attractors
  5. Invariant measure on the pullback attractors

Key Result

Theorem 2.1

Assume for each $\tau \in \mathbb{R}$, $f \in L_{l o c}^2(\mathbb{R} ; V^{\prime})$, $(u_\tau, \varphi) \in E_V^2$ and $g: \mathbb{R} \times C_V \rightarrow L^2(\Omega)$ satisfying (H1)-(H4). It is said that $u$ is a weak solution to 1.1 if and only if $u \in C([\tau,+\infty) ; V)$, $\frac{d u}{d t} for all $t \geq \tau$.

Theorems & Definitions (33)

  • Theorem 2.1
  • Theorem 2.2
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Definition 3.4
  • Lemma 3.1
  • Definition 3.5
  • Lemma 3.2
  • proof
  • ...and 23 more