Pullback exponential attractors for the three dimensional non-autonomous Navier-Stokes equations with nonlinear damping

Fang Li; Bo You

doi:10.3934/dcdsb.2019172

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2020, Volume 25, Issue 1: 55-80. Doi: 10.3934/dcdsb.2019172

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Pullback exponential attractors for the three dimensional non-autonomous Navier-Stokes equations with nonlinear damping

Fang Li^1, and
Bo You^2, ,

1.
School of Mathematics and Statistics, Xidian University, Xi'an 710126, China
2.
School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an 710049, China

^* Corresponding author: Bo You
^* Corresponding author: Bo You

Received: November 2018

Early access: July 2019

Published: January 2020

This work was supported by the National Science Foundation of China Grant (11401459, 11801427, 11871389), the Natural Science Foundation of Shaanxi Province (2018JQ1009, 2018JM1012) and the Fundamental Research Funds for the Central Universities (xjj2018088)

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Abstract

The main objective of this paper is to study the long-time behavior of solutions for the three dimensional non-autonomous Navier-Stokes equations with nonlinear damping for $ r>4. $ Inspired by the the methods of $ \ell $-trajectories in [27], we will prove the existence of a finite dimensional pullback attractor and a pullback exponential attractor, which gives another way of considering the long-time behavior of the non-autonomous evolutionary equations.

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Mathematics Subject Classification: Primary: 35B41, 37C60, 37L25; Secondary: 35Q30, 76D05.

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