Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Weak global attractor for the $3D$-Navier-Stokes equations via the globally modified Navier-Stokes equations

Matheus Cheque Bortolan, Alexandre Nolasco de Carvalho, Pedro Marín-Rubio, José Valero

Abstract

In this paper we obtain the existence of a weak global attractor for the three-dimensional Navier-Stokes equations, that is, a weakly compact set with an invariance property, that uniformly attracts solutions, with respect to the weak topology, for initial data in bounded sets. To that end, we define this weak global attractor in terms of limits of solutions of the globally modified Navier-Stokes equations in the weak topology. We use the theory of semilinear parabolic equations and $ε$-regularity to obtain the local well posedness for the globally modified Navier-Stokes equations and the existence of a global attractor and its regularity.

Weak global attractor for the $3D$-Navier-Stokes equations via the globally modified Navier-Stokes equations

Abstract

In this paper we obtain the existence of a weak global attractor for the three-dimensional Navier-Stokes equations, that is, a weakly compact set with an invariance property, that uniformly attracts solutions, with respect to the weak topology, for initial data in bounded sets. To that end, we define this weak global attractor in terms of limits of solutions of the globally modified Navier-Stokes equations in the weak topology. We use the theory of semilinear parabolic equations and -regularity to obtain the local well posedness for the globally modified Navier-Stokes equations and the existence of a global attractor and its regularity.
Paper Structure (8 sections, 23 theorems, 146 equations)

This paper contains 8 sections, 23 theorems, 146 equations.

Table of Contents

  1. Introduction
  2. Local existence and regularity results for abstract semilinear parabolic problems
  3. Regularity results
  4. Very weak formulation of the globally modified Navier-Stokes equation
  5. Nonlinearity of the GMNSE
  6. The very weak formulation of the GMNSE
  7. Global attractor in $H$ for the GMNSE
  8. Weak global attractors in H for the Navier-Stokes equation

Key Result

Theorem 1.1

Assume that $Pf\in H_{-\frac{1}{2}}$. For each $N>0$, equation NavierModified generates a semigroup $S_N=\{S_N(t)\colon t\geqslant 0\}$ in $H$ with a global attractor $\mathcal{A_N}$. For each $u_0\in H$, if $u(t) = S_N(t)u_0$ for $t\geqslant 0$, we have and $u$ is a classical solution of NavierModified. Moreover, for any $T>0$, $0<\eta<\frac{1}{8}$ and $p(\frac{1}{2}+\eta)<1$, In particular, fo

Theorems & Definitions (44)

  • Theorem 1.1
  • Definition 1.2
  • Theorem 1.3
  • Theorem 2.1: See AC
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • ...and 34 more