WORKSHOP
Analysis of Nematic Liquid Crystals Flows (2577)
Analyse des flux de cristaux liquides nématiques
Dates: 25-29 April 2022
Place: CIRM (Marseille Luminy, France)
DESCRIPTION

This workshop aims to
•  bring together leading experts in the field of liquid crystal flows to enhance dialog and discussions between the different groups working with the geometric or the fluid type approach,
•  stimulate new techniques for a better understanding of the general Ericksen and Leslie stress,
•  create analytical progress on the de Gennes Q-tensor model.

State of Research
The continuum theory of nematic liquid crystals was developed by Eriksen and Leslie in their pioneering work [6, 16] in the 1960s. Their theory models nematic liquid crystals flow from a hydrodynamic point of view and describes the evolution of the velocity of the fluid and the orientation of the rod-like liquid crystals. The original derivation of the model is based on the conservation laws for mass, momentum and angular momentum as well on constitutive relations given by Ericksen [6] and Leslie [16], which we nowadays call the Ericksen and Leslie stress. General liquid crystal materials are described by the Landau-de Gennes theory [5] from a unified point of view.
The authors in [11] showed that the Ericksen-Leslie model can be understood also from the perspective of thermodynamics and showed in particular that the full non-isothermal Ericksen-Leslie model, i.e., the case of a heat conducting, compressible fluid with general Leslie stress coupled with Ericksen’s anisotropic elasticity equations is thermodynamically consistent.
The rigorous analysis of the Ericksen-Leslie system began with the work of Lin [17] and Lin and Liu [18], who introduced and studied the so called simplified isothermal model. Two main approaches to the Ericksen-Leslie system have been developed since then: the geometric approach based on the theory of harmonic maps on spheres considering the director equation as master equation and the fluid approach considering the momentan balance equation as master equation, each having their advantages and  disadvantages.
Assuming Dirichlet boundary conditions Huang, Lin, Liu and Wang [15] were able to construct, using the geometric approach, examples of initial data (u0, d0) having sufficiently small energy and d0  fulfilling a topological condition for which the simplified model has  finite time blow up of (u, d).
On the other hand, assuming Neumann conditions a rather complete dynamic theory has been devel- oped in [10], which gives well-posedness result for the Ericksen-Leslie equations dealing with general Leslie stress without assuming additional conditions on the Leslie coefficients. For an extension to compressible situation see [12]. It is shown, that for initial data close to equilibria points (which are identical with the ones for the incompressible situation), the solution exists globally. Moreover, any global solution, which does not develop singularities, converges to an equilibrium in the topology of the natural state manifold. No structural assumptions on the Leslie coefficients are imposed and in particular, Parodi’s relation is not being assumed.
For further results concerning the incompressible isothermal case under various simplifications and various assumptions on the Leslie as well as Ericksen coefficients, we refer to [14], [20], [24], [25], [24], [23], [9], [19], to [7],[9],[10] for the non-isothermal case, as well to the recent survey article [11].
Another focal point of the workshop is the so called Q-tensor model introduced by de Gennes [5]. Here the molecular orientation is described by a function Q who takes values in the set of zero-trace symmetric matrices. Results for this model were pioneered by Ball [3], [2]. Further results in the weak or strong sense under various simplifications are due to Paicu and Zarnescu [21], Feireisl, Rocca, Schimperna and Zarnescu [8], Abels, Dolzmann and Liu [1] and [13].
Both models still lack a thorough analytical understanding. The workshop thus aims to enhance a better understanding of the following questions:
•  global (strong) weak well-posedness (for small data) of the Ericksen-Leslie model subject to general Leslie and general Ericksen stress,
•  determination of the set of equilibria, which depend strongly on the boundary conditions imposed,
•  convergence of solutions to an equilbrium set and long-time behaviour of solutions,
•  well-posedness of the Q-tensor model with nontrivial tunbling and alignment effects
SCIENTIFIC COMMITTEE
ORGANIZING COMMITTEE
SPEAKERS (list tba)
References
[1] H. Abels, G. Dolzmann, Y. Liu, Well-posedness of a fully coupled Navier-Stokes-Q-tensor system with inho- mogeneous boundary data. SIAM J. Math. Anal., 46 (2014), 3050-3077.
[2] J. Ball, Liquid crystal and their defects, Lect. Notes in Math. 2200, (2017), 1-46.
[3] J. Ball, A. Zarnescu, Orientability and energy minimization in liquid crystal models, Arch. Ration. Mech.
Anal. 202, (2011), 493-535.
[4] F. De Anna, C. Liu, Non-isothermal general Ericksen-Leslie system: derivation, analysis and thermodynamic consistency. Arch. Rational Mech. Anal. 231, (2019), 637-717.
[5] P.G. DeGennes, The Physics of Liquid Crystals, Oxford University Press, (1974).
[6] J. L. Ericksen, Hydrostatic theory of liquid crystals, Arch. Rational Mech. Anal. 9 (1962), 371–378.
[7] E. Feireisl, M. Fr´emond, E. Rocca, G. Schimperna, A new approach to non-isothermal models for nematic liquid crystals, Arch. Ration. Mech. Anal., 205, (2012), 651-672.
[8] E. Feireisl, E. Rocca, G. Schimperna, A. Zarnescu, Evolution of non-isothermal Landau-de Gennes nematic liquid crystal flows with singular potential,
[9] M. Hieber, M. Nesensohn, J. Pru¨ss, K. Schade, Dynamics of nematic lquid crystals: the quasilinear approach,
Ann. Inst. H. Poincar´e Anal. Non Lin´eaire,  33, (2016), 397-408.
[10]  M. Hieber, J. Pru¨ss, Dynamics of the Ericksen-Leslie equations with general Leslie stress I: The incompressible isotropic Case. Math. Ann., 369, (2017), 977-996.
[11] M. Hieber, J. Pru¨ss, Modeling and analysis of the Ericksen-Leslie equations for nematic liquid crystal flows. In: Handbook of Mathematical Analyis in Mechanics of Viscous Fluids, (Eds: Y. Giga, A. Novotny), 1075- 1134. Springer, 2018.
[12]  M. Hieber, J. Pru¨ss.  Dynamics of Ericksen-Leslie equations with general Leslie stress II: The compressible isotropic case.  Arch. Rational Mech. Anal., 233:1441-1468, 2019.
[13]  M. Hieber, M. Wrona, Well-posedness of the Q-tensor model with non-trivial tumbling and alignment effects, preprint.
[14]  M. Hong,  J. Li,  Z. Xin,  Blow-up criteria of strong solutions to the Ericksen-Leslie system in R3. Comm.
Partial Differential Equations, 39, (2014), 1284-1328.
[15]  T. Huang, F. Lin, C. Liu, C. Wang, Finite time singularities of the nematic liquid crystal flow in dimension three. arXiv:1504.01080v1.
[16] F. M. Leslie, Some constitutive equations for liquid crystals, Arch. Rational Mech. Anal. 28 (1968), 265–283.
[17]  F. Lin,  On  nematic liquid  crystals with  variable degree of  freedom,  Comm.  Pure  Appl.  Math.  44 (1991), 453-468.
[18] F. Lin, Ch. Liu, Nonparabolic dissipative systems modeling the flow of liquid crystals, Comm. Pure Appl.
Math. 48 (1995), 501–537.
[19]  F. Lin, C. Wang, Global existence of weak solutions of the nematic liquid crystal flow in dimension three.
Comm. Pure Appl. Math.   69, (2016), 1532-1571.
[20]  W. Ma, H. Gong, J. Li, Global strong solutions to incompressible Ericksen-Leslie system in R3. Nonlinear Anal., 109, (2014), 230-235.
[21] M. Paicu, A. Zarnescu, Global existence and regularity for full coupled Navier-Stokes and Q-tensor system.
SIAM J. Math. Anal., 43 (2011), 2009-2049.
[22] E. G. Virga, Variational Theories for Liquid Crystals, Chapman-Hall, London, 1994.
[23]  C. Wang, Well-posedness for the heat flow of harmonic maps and the liquid crystal flow with rough initial data, Arch. Ration. Mech. Anal. 200 (2011), 1-19.
[24]  W. Wang, P. Zhang, Z. Zhang, Well-posedness of the Ericksen-Leslie system, Arch. Ration. Mech. Anal. 210 (2013), 837-855.
[25]  H. Wu, X. Xu, Ch. Liu, On the general Ericksen-Leslie system:  Parodi’s relation, well-posedness and stability,
Arch. Ration. Mech. Anal., 208 (2013), 59-107.
SPONSORS OF THIS SEMESTER (to be completed)