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Space–Time Decay Estimates for the Incompressible Flow of Liquid Crystals

  • Qunyi Bie
  • Xuemei DengEmail author
  • Deyi Ma
Article

Abstract

In this paper, we study decay in space and time for derivatives of solutions to the incompressible flow of liquid crystals. Based on the known time decay results, we further obtain space–time decay rates agreed with those of the heat equation. More precisely, we firstly derive weighted bounds of the director field by weighted energy estimate, and then by regarding the nonlinear term \(\mathrm{div}{(}\nabla {d}\odot \nabla {d}{)}\) as a forcing term in the velocity equation, we could get the weighted bounds for the vorticity, which implies the corresponding estimates for the velocity field. Our results show that the weighted \(L^p\) decay for the velocity is not as good as that of both the director field and the vorticity. This is basically due to the presence of the pressure term in the velocity equation, which leads to the restriction on the exponent of the weight.

Keywords

Liquid crystal flow Space–time decay Weighted estimates 

Mathematics Subject Classification

35Q35 35K15 35B40 

Notes

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.College of Science and Three Gorges Mathematical Research CenterChina Three Gorges UniversityYichangPeople’s Republic of China

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