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Dynamics and flow effects in the Beris-Edwards system modeling nematic liquid crystals. (English) Zbl 1421.76016
The paper analyzes some properties of incompressible liquid crystal flows of nematic type. The authors consider the Beris-Edwards system which couples a Navier-Stokes system for the fluid velocity $$u:\mathbb{T}^{2}\times (0,+\infty )\rightarrow \mathbb{R}^{3}$$ and a parabolic reaction-convection-diffusion system for the tensor $$Q:\mathbb{T}^{2}\times (0,+\infty )\rightarrow \mathcal{S}_{0}^{(3)}$$ which describes the average orientation of liquid crystal molecules. Here, $$\mathbb{T}^{2}$$ is the 2D torus and $$\mathcal{S}_{0}^{(3)}=\{Q\in M^{3\times 3}\mid Q_{ij}=Q_{ji} \text{ for all } 1\leq i,j\leq 3,\,\mathrm{tr}(Q)=0\}$$. After some transformations, this Beris-Edwards system can be written as $\rho (u_{t}+u\cdot \nabla u)-\nu \Delta u+\nabla P=\nabla \cdot (\tau +\sigma ), \,\nabla \cdot u=0, \,Q_{t}+u\cdot \nabla Q-S(\nabla u,Q)=\Gamma H(Q).$ Initial conditions $$(u_{0},Q_{0})\in H^{1}(\mathbb{T}^{2};\mathbb{R}^{3})\times H^{2}(\mathbb{T} ^{2};\mathcal{S}_{0}^{(3)})$$ with $$\nabla \cdot u_{0}=0$$ are added to this Beris-Edwards system. The authors define the notion of global strong solution to this problem and they admit the existence of a global strong solution. In the co-rotational case, that is cancelling a non-dimensional constant measuring a characteristic of the molecules in $$S(\nabla u,Q)$$, the first main result proves that the eigenvalues of the solution $$Q$$ lie in the same interval at that of the initial $$Q_{0}$$, under restrictions on this interval. The second main result proves that, for every $$\varepsilon \in (0,1)$$, the global strong solution starting from $$(u_{0},Q_{0})\in H^{5}(\mathbb{T}^{2};\mathbb{R}^{3})\times H^{4}( \mathbb{T}^{2};\mathcal{S}_{0}^{(3)})$$ with $$\nabla \cdot u_{0}=0$$ is close to that $$(v,R)\in L^{\infty }(0,T;H^{5}(\mathbb{T}^{2};\mathbb{R} ^{3}))\times L^{\infty }(0,T;H^{4}(\mathbb{T}^{2};\mathcal{S}_{0}^{(3)})$$ of the limit problem
\begin{aligned} \partial _{t}v+v\cdot \nabla v+\nabla q &=0, \\ \nabla \cdot v &= 0,\\ \partial _{t}R+v\cdot \nabla R-\Omega _{v}R+R\Omega _{v} &=-aR+b(R^{2}- \frac{1}{3}tr(R^{2})\mathcal{I})-cRtr(R^{2}), \end{aligned}
where $$\Omega _{v}=\frac{ \nabla v-\nabla ^{T}v}{2}$$, and the authors give estimates on the difference $$w^{\varepsilon }=u^{\varepsilon }-v$$, $$S^{\varepsilon }=Q^{\varepsilon }-R$$. In the non co-rotational case, the authors prove a lack of eigenvalue preservation. For the proof of the preservation property, the authors rewrite the problem, introducing nondimensional variables, they analyze the flow generated by the ODE part of the $$Q$$-equation, introduce a linear non-autonomous problem and use maximum principles, approximations and passage to the limit with respect to these approximations. For the limit with respect to large Ericksen number, the authors analyze the equation satisfied by $$w^{\varepsilon }=u^{\varepsilon }-v$$. In the last part, they analyze the above-indicated limit system, and they especially follow the occurrence of high gradients of $$R$$, called defects, considering in few examples two distinct ways leading to defects: phases mismatch or vorticity-driven defects.

##### MSC:
 76A15 Liquid crystals 35Q35 PDEs in connection with fluid mechanics
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##### References:
 [1] Abels, H.; Dolzmann, G.; Liu, YN, Well-posedness of a fully coupled Navier-Stokes/Q-tensor system with inhomogeneous boundary data, SIAM J. Math. Anal., 46, 3050-3077, (2014) · Zbl 1336.35283 [2] Abels, H.; Dolzmann, G.; Liu, YN, Strong solutions for the Beris-Edwards model for nematic liquid crystals with homogeneous Dirichlet boundary conditions, Adv. Differ. Equ., 21, 109-152, (2016) · Zbl 1333.35174 [3] Ball, J., Differentiability properties of symmetric and isotropic functions, Duke Math. J., 51, 699-728, (1984) · Zbl 0566.73001 [4] Ball, J.: Mathematics of Liquid Crystals. Cambridge Centre for Analysis, Short Course Slides, 13-17, 2012 [5] Ball, J.; Majumdar, A., Nematic liquid crystals: from Maier-Saupe to a continuum theory, Mol. Cryst. Liq. Cryst., 525, 1-11, (2010) [6] Barberi, R., Ciuchi, F., Durand, G.E., Iovane, M., Sikharulidze, D., Sonnet, A.M., Virga, E.G.: Electric field induced order reconstruction in a nematic cell. Eur. Phys. J. E Soft Matter Biol. Phys.13(1), 61-71 (2004) [7] Bauman, P.; Phillips, D., Regularity and the behavior of eigenvalues for minimizers of a constrained Q-tensor energy for liquid crystals, Calc. Var. Partial Differ. Equ., 55, 55-81, (2016) · Zbl 1401.35068 [8] Beris, A.N., Edwards, B.J.: Thermodynamics of Flowing Systems with Internal Microstructure. Oxford Engineerin Science Series, vol. 36. Oxford university Press, Oxford, New York, 1994 [9] Boyd, S.; Vandenberghe, L.: Convex Optimization. Cambridge University Press, Cambridge (2004) · Zbl 1058.90049 [10] Cavaterra, C.; Rocca, E.; Wu, H.; Xu, X., 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, 1368-1399, (2016) · Zbl 1348.35166 [11] Dai, MM; Feireisl, E.; Rocca, E.; Schimperna, G.; Schonbek, M., On asymptotic isotropy for a hydrodynamic model of liquid crystals, Asymptot. Anal., 97, 189-210, (2016) · Zbl 1345.35073 [12] Anna, F., A global 2D well-posedness result on the order tensor liquid crystal theory, J. Differ. Equ., 262, 3932-3979, (2017) · Zbl 1355.76010 [13] Anna, F.; Zarnescu, A., Uniqueness of weak solutions of the full coupled Navier-Stokes and Q-tensor system in 2D, Commun. Math. Sci., 14, 2127-2178, (2016) · Zbl 1355.35147 [14] Fefferman, CL; McCormick, DS; Robinson, JC; Rodrigo, JL, Higher order commutator estimates and local existence for the non-resistive MHD equations and related models, J. Funct. Anal., 267, 1035-1056, (2014) · Zbl 1296.35142 [15] de Gennes, P.G.; Prost, J.: The Physics of Liquid Crystals. Oxford Science Publications, Oxford (1993) [16] Evans, LC; Kneuss, O.; Tran, H., Partial regularity for minimizers of singular energy functionals, with application to liquid crystal models, Trans. Am. Math. Soc., 368, 3389-3413, (2016) · Zbl 1337.49067 [17] Feireisl, E.; Rocca, E., Schimperna, G., arnescu, A.: Evolution of non-isothermal Landau-de Gennes nematic liquid crystals flows with singular potential, Commun. Math. Sci., 12, 317-343, (2014) · Zbl 1330.76012 [18] Guckenheimer, J., Holmes, P.J. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, vol. 42, Springer Science & Business Media, 2013 · Zbl 0515.34001 [19] Guillén-González, F.; Rodríguez-Bellido, MA, Weak time regularity and uniqueness for a Q-tensor model, SIAM J. Math. Anal., 46, 3540-3567, (2014) · Zbl 1315.35167 [20] Guillén-González, F.; Rodríguez-Bellido, MA, Weak solutions for an initial-boundary Q-tensor problem related to liquid crystals, Nonlinear Anal., 112, 84-104, (2015) · Zbl 1304.35547 [21] Feireisl, E.; Rocca, E.; Schimperna, G.; Zarnescu, A., Nonisothermal nematic liquid crystal flows with the Ball-Majumdar free energy, Annali di Mat. Pura ed App., 194, 1269-1299, (2015) · Zbl 1446.76067 [22] Hartman, P.: Ordinary Differential Equations. Reprint of the second edition, Birkhäuser, Boston, MA (1982) · Zbl 0476.34002 [23] Ionescu, A.D., Kenig, C.E.: Local and global well-posedness of periodic KP-I equations. Mathematical Aspects of Nonlinear Dispersive Equations. Annals of Mathematics Studies, Vol. 163, Princeton University Press, 181-212, 2009 [24] Iyer, G.; Xu, X.; Zarnescu, A., Dynamic cubic instability in a 2D Q-tensor model for liquid crystals, Math. Models Methods Appl. Sci., 25, 1477-1517, (2015) · Zbl 1328.82053 [25] Kato, T.; Ponce, G., Commutator estimates and the Euler and Navier-Stokes equations, Commun. Pure Appl. Math., 41, 891-907, (1988) · Zbl 0671.35066 [26] Liu, C.; Calderer, MC, Liquid crystal flow: dynamic and static configurations, SIAM J. Appl. Math., 60, 1925-1949, (2000) · Zbl 0956.35104 [27] Mottram, N.J., Newton, J.P.: Introduction to Q-tensor theory. Preprint, arXiv:1409.3542, 2014 [28] Majda, A.J.: Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Applied Mathematical Sciences, vol. 53. Springer-Verlag, New York (1984) · Zbl 0537.76001 [29] Majda, A.J., Bertozzi, A.L.: Vorticity and Incompressible Flow, vol. 27. Cambridge University Press, 2002 · Zbl 0983.76001 [30] Majumdar, A., Equilibrium order parameters of nematic liquid crystals in the Landau-de Gennes theory, Eur. J. Appl. Math., 21, 181-203, (2010) · Zbl 1253.82089 [31] Majumdar, A.; Zarnescu, A., Landau-De Gennes theory of nematic liquid crystals: the Oseen-Frank limit and beyond, Arch. Ration. Mech. Anal., 196, 227-280, (2010) · Zbl 1304.76007 [32] Nomizu, K., Characteristic roots and vectors of a diifferentiable family of symmetric matrices, Linear Multilinear Algebra, 1, 159-162, (1973) · Zbl 0277.15004 [33] Paicu, M.; Zarnescu, A., Global existence and regularity for the full coupled Navier-Stokes and Q-tensor system, SIAM J. Math. Anal., 43, 2009-2049, (2011) · Zbl 1233.35160 [34] Paicu, M.; Zarnescu, A., Energy dissipation and regularity for a coupled Navier-Stokes and Q-tensor system, Arch. Ration. Mech. Anal., 203, 45-67, (2012) · Zbl 1315.76017 [35] Mkaddem, S.; Gartland, EC, Fine structure of defects in radial nematic droplets, Phys. Rev. E, 62, 6694, (2000) [36] Murza, AC; Teruel, AE; Zarnescu, A., Shear flow dynamics in the Beris-Edwards model of nematic liquid crystals, Proc. R. Soc. A, 474, 20170673, (2018) · Zbl 1402.76022 [37] Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, vol. 44. Springer-Verlag, New York (1983) · Zbl 0516.47023 [38] Taylor, M.E.: Partial Differential Equations. III. Nonlinear Equations, Applied Mathematical Sciences, vol. 117, Springer-Verlag, New York, 1997 [39] Wilkinson, M., Strict physicality of global weak solutions of a Navier-Stokes Q-tensor system with singular potential, Arch. Ration. Mech. Anal., 218, 487-526, (2015) · Zbl 1329.35255
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