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Suitable Weak Solutions for the Co-rotational Beris–Edwards System in Dimension Three

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Abstract

In this paper, we establish the global existence of a suitable weak solution to the co-rotational Beris–Edwards Q-tensor system modeling the hydrodynamic motion of nematic liquid crystals with either Landau–De Gennes bulk potential in \({\mathbb {R}}^3\) or Ball–Majumdar bulk potential in \(\mathbb {T}^3\), a system coupling the forced incompressible Navier–Stokes equation with a dissipative, parabolic system of Q-tensor Q in \({\mathbb {R}}^3\), which is shown to be smooth away from a closed set \(\Sigma \) whose 1-dimensional parabolic Hausdorff measure is zero.

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Notes

  1. Strictly speaking, we need to take finite quotient \(D_h^j\) of (1.6) \((j=1,2,3\)) and then send \(h\rightarrow 0\).

  2. Strictly speaking, we need to multiply \(\Delta (D^j_h Q)\eta ^2\) and \(\nabla (D_h^j\mathbf{u})\eta ^2\) and then send \(h\rightarrow 0\).

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Acknowledgements

Both the first and third authors are partially supported by NSF Grant 1764417. The second author is partially supported by the GRF grant (Project No. CityU 11332216). The third author wishes to express his gratitude to Professor Fanghua Lin for helpful discussions related to the blowing up argument in this paper.

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Authors and Affiliations

  1. Department of Mathematics, Purdue University, West Lafayette, IN, 47907, USA

    Hengrong Du & Changyou Wang

  2. Department of Mathematics, City University of Hong Kong, Hong Kong, People’s Republic of China

    Xianpeng Hu

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  1. Hengrong Du
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  2. Xianpeng Hu
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Correspondence to Changyou Wang.

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Communicated by F. Lin.

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Du, H., Hu, X. & Wang, C. Suitable Weak Solutions for the Co-rotational Beris–Edwards System in Dimension Three. Arch Rational Mech Anal 238, 749–803 (2020). https://doi.org/10.1007/s00205-020-01554-y

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