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[J]. Electronic Research Archive, 2021, 29(6): 3817-3832. doi: 10.3934/era.2021063
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.
[1] | Well-posedness of a fully-coupled Navier-Stokes/Q-tensor system with inhomogeneous boundary data. SIAM J. Math. Anal. (2014) 46: 3050-3077. |
[2] | Strong solutions for the Beris-Edwards model for nematic liquid crystals with homogeneous Dirichlet boundary conditions. Adv. Differential Equations (2016) 21: 109-153. |
[3] | Nematic liquid crystals: From Maier-Saupe to a continuum theory. Molecular Crystals and Liquid Crystals (2010) 525: 1-11. |
[4] | A. N. Beris and B. J. Edwards, Thermodynamics of Flowing Systems with Internal Microstructure, Oxford University Press: New York, 1994. |
[5] | Biphasic, lyotropic, active nematics. Phys. Rev. Lett. (2014) 113: 248-303. |
[6] | Global strong solutions of the full Navier-Stokes and $Q$-tensor system for nematic liquid crystal flows in two dimensions. SIAM J. Math. Anal. (2016) 48: 1368-1399. |
[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. |
[8] | Global existence and regularity of solutions for the active liquid crystals. J. Differential Equation (2017) 263: 202-239. |
[9] | Global weak solutions for the compressible active liquid crystal system. SIAM J. Math. Anal. (2018) 50: 3632-3675. |
[10] | Dynamics of bacterial swarming. Biophys. J. (2010) 98: 2082-2090. |
[11] | Uniqueness of weak solutions of the full coupled Navier-Stokes and $Q$-tensor system in 2D. Commun. Math. Sci. (2016) 14: 2127-2178. |
[12] | P. G. de Gennes and J. Prost, The Physics of Liquid Crystals, Second Edition, Oxford University Press: New York, 1995. |
[13] | M. Doi and S. F. Edwards, The Theory of Polymer Dynamics, Oxford University Press: New York, 1986. |
[14] | Suitable weak solutions for the co-rotational Beris-Edwards system in dimension three. Arch. Ration. Mech. Anal. (2020) 238: 749-803. |
[15] | Conservation laws for liquid crystals. Trans. Soc. Rheology. (1961) 5: 23-34. |
[16] | (2004) Dynamics of Viscous Compressible Fluids.The Clarendon Press. |
[17] | On the existence of globally defined weak solutions to the Navier-Stokes equations. J. Math. Fluid Mech. (2001) 3: 358-392. |
[18] | Evolution of non-isothermal Landau-de Gennes nematic liquid crystals flows with singular potential. Commun. Math. Sci. (2014) 12: 317-343. |
[19] | Nonisothermal nematic liquid crystal flows with the Ball-Majumdar free energy. Ann. Mat. Pura Appl. (2015) 194: 1269-1299. |
[20] | S. M. Fielding, D. Marenduzzo and M. E. Cates, Nonlinear dynamics and rheology of active fluids: Simulations in two dimensions, Phys. Rev. E., 83 (2011), 041910. doi: 10.1103/PhysRevE.83.041910 |
[21] | On the theory of liquid crystals. Discussions Faraday Soc. (1958) 25: 19-28. |
[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 |
[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 |
[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 |
[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 |
[26] | Banding, excitability and chaos in active nematic suspensions. Nonlinearity (2012) 25: 2245-2269. |
[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 |
[28] | Weak time regularity and uniqueness for a $Q$-tensor model. SIAM J. Math. Anal. (2014) 46: 3540-3567. |
[29] | Weak solutions for an initial-boundary Q-tensor problem related to liquid crystals. Nonlinear Anal. (2015) 112: 84-104. |
[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). |
[31] | Global solution to the 3D inhomogeneous nematic liquid crystal flows with variable density. J. Differential Equations (2018) 264: 5300-5332. |
[32] | Global solution to the three-dimensional incompressible flow of liquid crystals. Comm. Math. Phys. (2010) 296: 861-880. |
[33] | Global solution to the three-dimensional compressible flow of liquid crystals. SIAM J. Math. Anal. (2013) 45: 2678-2699. |
[34] | Global well-posedness for the dynamical $Q$-tensor model of liquid crystals. Sci. China Math. (2015) 58: 1349-1366. |
[35] | On multi-dimensional compressible flows of nematic liquid crystals with large initial energy in a bounded domain. J. Funct. Anal. (2013) 265: 3369-3397. |
[36] | Global weak solutions to the equations of compressible flow of nematic liquid crystals in two dimensions. Arch. Ration. Mech. Anal. (2014) 214: 403-451. |
[37] | Some constitutive equations for liquid crystals. Arch. Ration. Mech. Anal. (1968) 28: 265-283. |
[38] | Global strong solution to the density-dependent incompressible flow of liquid crystals. Trans. Amer. Math. Soc. (2015) 367: 2301-2338. |
[39] | Global weak solutions to the active hydrodynamics of liquid crystals. J. Differential Equations (2020) 268: 4194-4221. |
[40] | Nonparabolic dissipative systems modeling the flow of liquid crystals. Comm. Pure Appl. Math. (1995) 48: 501-537. |
[41] | Static and dynamic theories of liquid crystals.. J. Partial Differential Equations (2001) 14: 289-330. |
[42] | F. Lin and C. Wang, Recent developments of analysis for hydrodynamic flow of nematic liquid crystals, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 372 (2014), 20130361, 18 pp. doi: 10.1098/rsta.2013.0361 |
[43] | Global existence of weak solutions of the nematic liquid crystal flow in dimension three. Comm. Pure Appl. Math. (2016) 69: 1532-1571. |
[44] | Q. Liu, C. Wang, X. Zhang and J. Zhou, On optimal boundary control of Ericksen-Leslie system in dimension two, Calc. Var. Partial Differential Equations, 59 (2020), Paper No. 38, 64 pp. doi: 10.1007/s00526-019-1676-z |
[45] | D. Marenduzzo, E. Orlandini, M. E. Cates and J. M. Yeomans, Steady-state hydrodynamic instabilities of active liquid crystals: Hybrid lattice Boltzmann simulations, Phys. Rev. E., 76 (2007), 031921. doi: 10.1103/PhysRevE.76.031921 |
[46] | S. Mishra, A. Baskaran and M. C. Marchetti, Fluctuations and pattern formation in self-propelled particles, Phys. Rev. E., 81 (2010), 061916. doi: 10.1103/PhysRevE.81.061916 |
[47] | S. Mishra and S. Ramaswamy, Active nematics are intrinsically phase separated, Phys. Rev. Lett., 97 (2006), 090602. doi: 10.1103/PhysRevLett.97.090602 |
[48] | Long-lived giant number fluctuations in a swarming granular nematic. Science (2007) 317: 105-108. |
[49] | The theory of liquid crystals. Trans. Faraday Soc. (1933) 29: 883-899. |
[50] | Global existence and regularity for the full coupled Navier-Stokes and $Q$-tensor system. SIAM J. Math. Anal. (2011) 43: 2009-2049. |
[51] | Energy dissipation and regularity for a coupled Navier-Stokes and $Q$-tensor system. Arch. Ration. Mech. Anal. (2012) 203: 45-67. |
[52] | Hydrodynamic phenomena in suspensions of swimming microorganisms. Annu. Rev. Fluid Mech. (1992) 24: 313-358. |
[53] | Martingale solution for stochastic active liquid crystal system. Discrete and Continuous Dynamical Systems (2021) 41: 2227-2268. |
[54] | Z. Qiu and Y. Wang, Strong solution for compressible liquid crystal system with random force, Submitted, (2020), arXiv: 2003.06074 [math.AP]. |
[55] | The mechanics and statistics of active matter. Annu. Rev. Condens. Matter Phys. (2010) 1: 323-345. |
[56] | Active nematics on a substrate: Giant number fluctuations and long-time tails. Europhys. Lett. (2003) 62: 196-202. |
[57] | M. Ravnik and J. M. Yeomans, Confined active nematic flow in cylindrical capillaries, Phys. Rev. Lett., 110 (2013), 026001. doi: 10.1103/PhysRevLett.110.026001 |
[58] | D. Saintillan and M. J. Shelley, Instabilities and pattern formation in active particle suspensions: Kinetic theory and continuum simulations, Phys. Rev. Lett., 100 (2008), 178103. doi: 10.1103/PhysRevLett.100.178103 |
[59] | Spontaneous motion in hierarchically assembled active matter. Nature. (2012) 491: 431-434. |
[60] | A. Sokolov and I. S. Aranson, Reduction of viscosity in suspension of swimming bacteria, Phys. Rev. Lett., 103 (2009), 148101. doi: 10.1103/PhysRevLett.103.148101 |
[61] | R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, Reprint of the 1984 Edition. AMS Chelsea Publishing: Providence, RI, 2001. doi: 10.1090/chel/343 |
[62] | E. G. Virga, Variational Theories for Liquid Crystals, Chapman & Hall: London, 1994. |
[63] | R. Voituriez, J. F. Joanny and J. Prost, Spontaneous flow transition in active polar gels, Europhys. Lett., 70 (2005), 404. doi: 10.1209/epl/i2004-10501-2 |
[64] | Global weak solution for a coupled compressible Navier-Stokes and Q-tensor system. Commun. Math. Sci. (2015) 13: 49-82. |
[65] | Global weak solution and large-time behavior for the compressible flow of liquid crystals. Arch. Ration. Mech. Anal. (2012) 204: 881-915. |
[66] | Incompressible limit for the compressible flow of liquid crystals. J. Math. Fluid Mech. (2014) 16: 771-786. |
[67] | Y. Wang, Incompressible limit of the compressible $Q$-tensor system of liquid crystals, Preprint. |
[68] | Meso-scale turbulence in living fluids. Proc. Natl. Acad. Sci. (2012) 109: 14308-14313. |
[69] | Strict physicality of global weak solutions of a Navier-Stokes $Q$-tensor system with singular potential. Arch. Rational Mech. Anal. (2015) 218: 487-526. |
[70] | A. Zarnescu, Mathematical problems of nematic liquid crystals: Between dynamical and stationary problems, Phil. Trans. R. Soc. A, 379 (2021), 20200432. doi: 10.1098/rsta.2020.0432 |