Dynamics of Partial Differential Equations

Volume 14 (2017)

Number 2

Backwards compact attractors for non-autonomous damped 3D Navier–Stokes equations

Pages: 201 – 218

DOI: https://dx.doi.org/10.4310/DPDE.2017.v14.n2.a4

Authors

Jinyan Yin (School of Mathematics and Statistics, Southwest University, Chongqing, China)

Anhui Gu (School of Mathematics and Statistics, Southwest University, Chongqing, China)

Yangrong Li (School of Mathematics and Statistics, Southwest University, Chongqing, China)

Abstract

Both existence and backwards topological property of pullback attractors are discussed for 3D Navier-Stokes equations with a nonlinear damping and a non-autonomous force. A pullback attractor is obtained in a square integrable space if the order of damping is larger than three and further in a Sobolev space if the order belongs to $(3, 5)$, the latter of which improves the best range $[7/2, 5)$ given in literatures so far. The new hypotheses on the force used here are weaker than those given in literatures. More importantly, the obtained attractor is shown to be backwards compact, i.e. the union of attractors over the past time is pre-compact. This result is a successful application of some new abstract criteria on backwards compact attractors if an evolution process is backwards pullback limit-set compact or equivalently backwards pullback flattening.

Keywords

backwards compactness, pullback attractor, non-autonomous system, 3D Navier–Stokes equation, nonlinear damping

2010 Mathematics Subject Classification

35B40, 35B41, 37L05

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Published 31 May 2017