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Dynamics of 2D Incompressible Non-autonomous Navier–Stokes Equations on Lipschitz-like Domains

  • Xin-Guang Yang
  • Yuming QinEmail author
  • Yongjin Lu
  • To Fu Ma
Article
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Abstract

This paper concerns the tempered pullback dynamics of 2D incompressible non-autonomous Navier–Stokes equations with a non-homogeneous boundary condition on Lipschitz-like domains. With the presence of a time-dependent external force f(t) which only needs to be pullback translation bounded, we establish the existence of a minimal pullback attractor with respect to a universe of tempered sets for the corresponding non-autonomous dynamical system. We then give estimates on the finite fractal dimension of the attractor based on trace formula. Under the additional assumption that the external force is perturbed from a stationary force by a time-dependent perturbation, we also prove the upper semi-continuity of the attractors as the non-autonomous perturbation vanishes. Lastly, we investigate the regularity of these attractors when smoother initial data are given. Our results are new even for smooth domains.

Keywords

Navier–Stokes equation Lipshitz-like domain Universe Pullback tempered Pullback translation bounded Attractors 

Mathematics Subject Classification

35Q30 35B40 35B41 76D03 76D05 

Notes

Acknowledgements

This work was initiated when Xin-Guang Yang was a long term visitor as Posdocotor at ICMC-USP, Brazil, from May 2015 to June 2016, supported by FAPESP (Grant No. 2014/17080-0). He was also partially supported by NSFC of China (Grant No.11726626) and the Fund of Young Backbone Teacher in Henan Province (No. 2018GGJS039). Yongjin Lu was partially supported by NSF (Grant No. 1601127), they were also supported by the Key Project of Science and Technology of Henan Province (Grant No. 182102410069). Yuming Qin was in part supported by the NSFC of China (Grant No. 11671075). T. F. Ma was partially supported by CNPq (Grant No. 310041/2015-5).

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Authors and Affiliations

  1. 1.Department of Mathematics and Information ScienceHenan Normal UniversityXinxiangPeople’s Republic of China
  2. 2.Department of Applied MathematicsDonghua UniversityShanghaiPeople’s Republic of China
  3. 3.Department of Mathematics and EconomicsVirginia State UniversityPetersburgUSA
  4. 4.Institute of Mathematical and Computer SciencesUniversity of São PauloSão CarlosBrazil

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