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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) |
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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)
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SPONSORS OF THIS SEMESTER (to be completed)
