Global attractors for weak solutions of the three-dimensional Navier-Stokes equations with damping

Daniel Pardo; José Valero; Ángel Giménez

doi:10.3934/dcdsb.2018279

Article Contents

2019, Volume 24, Issue 8: 3569-3590. Doi: 10.3934/dcdsb.2018279

This issue Previous Article On asymptotically autonomous dynamics for multivalued evolution problems Next Article Modeling and analysis of random and stochastic input flows in the chemostat model

Global attractors for weak solutions of the three-dimensional Navier-Stokes equations with damping

1.
Miquel Rosselló i Alemany, 51, 7 B, Palma de Mallorca, Spain
2.
Universidad Miguel Hernández de Elche, Centro de Investigación Operativa, 03202, Elche (Alicante), Spain

* Corresponding author: jvalero@umh.es
* Corresponding author: jvalero@umh.es

Dedicated to Professor Peter Kloeden on his 70th birthday

Received: February 28, 2018

Revised: May 31, 2018

Early access: October 2018

Published: July 2019

This work has been partially supported by Spanish Ministry of Economy and Competitiveness and FEDER, projects MTM2015-63723-P and MTM2016-74921-P, and by Junta de Andalucía (Spain), project P12-FQM-1492

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Abstract

In this paper we obtain the existence of global attractors for the dynamicalsystems generated by weak solution of the three-dimensional Navier-Stokesequations with damping. We consider two cases, depending on the values of the parameter β controlling the damping term. First, we prove that for β≥4 weaksolutions are unique and establish the existence of the global attractor forthe corresponding semigroup. Second, for 3≤β<4 we define amultivalued dynamical systems and prove the existence of the global attractoras well. Finally, some numerical simulations are performed.

Keywords:

Three-dimensional Navier-Stokes equations with damping,
global attractors,
set-valued dynamical systems,
asymptotic behaviour,
turbulence.

Mathematics Subject Classification: 35B40, 35B41, 35K55, 35Q30, 37B25, 58C06.

Citation:

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Figure 1. Flow domain

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Figure 2. Flow velocity $u$ for the steady state in the $xy$-section at $z = 0$. The darker areas mean lower fluid flow speed. The initial velocity $u_{0}$ is identically zero. Left panel parameters: $\alpha = 0.2;\, \beta = 1$. Right panel parameters: $\alpha = 0.5;\, \beta = 1$

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Figure 3. Flow velocity $u$ for the steady state in the $xy$-section at $z = 0$. The darker areas mean lower fluid flow speed. The initial velocity $u_{0}$ is identically zero. Left panel parameters: $\alpha = 0.2;\, \beta = 2$. Right panel parameters: $\alpha = 0.5;\, \beta = 2$

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Figure 4. Flow velocity $u$ for the steady state in the $xy$-section at $z = 0$. The darker areas mean lower fluid flow speed. The initial velocity $u_{0}$ is identically zero. Left panel parameters: $\alpha = 0.2;\, \beta = 4$. Right panel parameters: $\alpha = 0.5;\, \beta = 4$

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Figure 5. Flow velocity $u$ in the $xy$-section at $z = 0$ for $\alpha = 0.2$ and $\beta = 1$. The darker areas mean lower fluid flow speed. The initial velocity $u_{0} = (1, 0, 0)$. Left panel: state when $t = 0.1$. Right panel: steady state ($t$ large enough)

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