doi: 10.3934/era.2021063
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On mathematical analysis of complex fluids in active hydrodynamics

1. 

College of Mathematics and Physics, Beijing University of Chemical Technology, Beijing 100029, China

2. 

Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, USA

* Corresponding author: Dehua Wang

Received  May 2021 Revised  June 2021 Early access August 2021

Fund Project: The research of Y. Chen was supported in part by National Natural Sciences Foundation of China No.11671027, 11901025. The research of D. Wang was supported in part by the National Science Foundation under grant DMS-1907519. The research of R. Zhang was supported in part by the National Science Foundation under grant DMS-1613213 and DMS-1613375

This is a survey article for this special issue providing a review of the recent results in the mathematical analysis of active hydrodynamics. Both the incompressible and compressible models are discussed for the active liquid crystals in the Landau-de Gennes Q-tensor framework. The mathematical results on the weak solutions, regularity, and weak-strong uniqueness are presented for the incompressible flows. The global existence of weak solution to the compressible flows is recalled. Other related results on the inhomogeneous flows, incompressible limits, and stochastic analysis are also reviewed.

Citation: Yazhou Chen, Dehua Wang, Rongfang Zhang. On mathematical analysis of complex fluids in active hydrodynamics. Electronic Research Archive, doi: 10.3934/era.2021063
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show all references

References:
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H. AbelsG. Dolzmann and Y.-N. Liu, Well-posedness of a fully-coupled Navier-Stokes/Q-tensor system with inhomogeneous boundary data, SIAM J. Math. Anal., 46 (2014), 3050-3077.  doi: 10.1137/130945405.  Google Scholar

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M. L. BlowS. P. Thampi and J. M. Yeomans, Biphasic, lyotropic, active nematics, Phys. Rev. Lett., 113 (2014), 248-303.  doi: 10.1103/PhysRevLett.113.248303.  Google Scholar

[6]

C. CavaterraE. RoccaH. Wu and X. Xu, Global strong solutions of the full Navier-Stokes and $Q$-tensor system for nematic liquid crystal flows in two dimensions, SIAM J. Math. Anal., 48 (2016), 1368-1399.  doi: 10.1137/15M1048550.  Google Scholar

[7]

H. Chaté, F. Ginelli and R. Montagne, Simple model for active nematics: Quasi-long-range order and giant fluctuations, Phys. Rev. Lett., 96 (2006), 180602. Google Scholar

[8]

G.-Q. ChenA. MajumdarD. Wang and R. Zhang, Global existence and regularity of solutions for the active liquid crystals, J. Differential Equation, 263 (2017), 202-239.  doi: 10.1016/j.jde.2017.02.035.  Google Scholar

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[21]

F. C. Frank, On the theory of liquid crystals, Discussions Faraday Soc., 25 (1958), 19-28.   Google Scholar

[22]

F. Ginelli, F. Peruani, M. Bär and H. Chaté, Large-scale collective properties of self-propelled rods, Phys. Rev. Lett., 104 (2010), 184502. doi: 10.1103/PhysRevLett.104.184502.  Google Scholar

[23]

L. Giomi, M. J. Bowick, X. Ma and M. C. Marchetti, Defect annihilation and proliferation in active nematics, Phys. Review Lett., 110 (2013), 228101. doi: 10.1103/PhysRevLett.110.228101.  Google Scholar

[24]

L. Giomi, T. B. Liverpool and M. C. Marchetti, Sheared active fluids: Thickening, thinning, and vanishing viscosity, Phys. Rev. E., 81 (2010), 051908, 9 pp. doi: 10.1103/PhysRevE.81.051908.  Google Scholar

[25]

L. Giomi, L. Mahadevan, B. Chakraborty and M. F. Hagan, Excitable patterns in active nematics, Phys. Rev. Lett., 106 (2011), 218101. doi: 10.1103/PhysRevLett.106.218101.  Google Scholar

[26]

L. GiomiL. MahadevanB. Chakraborty and M. F. Hagan, Banding, excitability and chaos in active nematic suspensions, Nonlinearity, 25 (2012), 2245-2269.  doi: 10.1088/0951-7715/25/8/2245.  Google Scholar

[27]

L. Giomi, M. C. Marchetti and T. B. Liverpool, Complex spontaneous flows and concentration banding in active polar films, Phys. Rev. Lett., 101 (2008), 198101. doi: 10.1103/PhysRevLett.101.198101.  Google Scholar

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F. Guillén-González and M. Rodriguez-Bellido, Weak time regularity and uniqueness for a $Q$-tensor model, SIAM J. Math. Anal., 46 (2014), 3540-3567.   Google Scholar

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F. Guill{é}n-Gonz{á}lez and M. Rodriguez-Bellido, Weak solutions for an initial-boundary Q-tensor problem related to liquid crystals, Nonlinear Anal., 112 (2015), 84-104.  doi: 10.1016/j.na.2014.09.011.  Google Scholar

[30]

M. Hieber and J. W. Prüss, Modeling and analysis of the Ericksen-Leslie equations for nematic liquid crystal flows, Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, 1075–1134, Springer, Cham, (2018).  Google Scholar

[31]

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[32]

X. Hu and D. Wang, Global solution to the three-dimensional incompressible flow of liquid crystals, Comm. Math. Phys., 296 (2010), 861-880.  doi: 10.1007/s00220-010-1017-8.  Google Scholar

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[36]

F. JiangS. Jiang and D. Wang, Global weak solutions to the equations of compressible flow of nematic liquid crystals in two dimensions, Arch. Ration. Mech. Anal., 214 (2014), 403-451.  doi: 10.1007/s00205-014-0768-3.  Google Scholar

[37]

F. M. Leslie, Some constitutive equations for liquid crystals, Arch. Ration. Mech. Anal., 28 (1968), 265-283.  doi: 10.1007/BF00251810.  Google Scholar

[38]

X. Li and D. Wang, Global strong solution to the density-dependent incompressible flow of liquid crystals, Trans. Amer. Math. Soc., 367 (2015), 2301-2338.  doi: 10.1090/S0002-9947-2014-05924-2.  Google Scholar

[39]

W. Lian and R. Zhang, Global weak solutions to the active hydrodynamics of liquid crystals, J. Differential Equations, 268 (2020), 4194-4221.  doi: 10.1016/j.jde.2019.10.020.  Google Scholar

[40]

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