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From kinetic to fluid models of liquid crystals by the moment method

  • * Corresponding author: Pierre Degond

    * Corresponding author: Pierre Degond 

In memory of Bob Glassey
PD is a visiting professor of the Department of Mathematics, Imperial College London, UK
AF acknowledges support from the Project EFI ANR-17-CE40-0030 of the French National Research Agency
JGL acknowledges support from the Department of Mathematics, Imperial College London, under Nelder Fellowship award and the National Science Foundation under Grants DMS-1812573 and DMS-2106988

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  • This paper deals with the convergence of the Doi-Navier-Stokes model of liquid crystals to the Ericksen-Leslie model in the limit of the Deborah number tending to zero. While the literature has investigated this problem by means of the Hilbert expansion method, we develop the moment method, i.e. a method that exploits conservation relations obeyed by the collision operator. These are non-classical conservation relations which are associated with a new concept, that of Generalized Collision Invariant (GCI). In this paper, we develop the GCI concept and relate it to geometrical and analytical structures of the collision operator. Then, the derivation of the limit model using the GCI is performed in an arbitrary number of spatial dimensions and with non-constant and non-uniform polymer density. This non-uniformity generates new terms in the Ericksen-Leslie model.

    Mathematics Subject Classification: 35Q35, 76A10, 76A15, 82C22, 82C70, 82D30, 82D60.


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  • Figure 1.  Graphical representation of the function $ \lambda \mapsto \rho^n(\lambda) $ (after [56]). (a) case $ n = 2 $. (b) case $ n \geq 3 $. The portions of the curves that correspond to stable equilibria are in blue, the unstable ones, in green.

    Figure 2.  Graphical representation of Condition (101). The ambient three-dimensional space in the figure represents the flat space $ {\mathcal S}_0^n $ in which $ {\mathcal U}_0^n $ is an imbedded manifold represented by a surface. $ {\mathcal N} $ is a submanifold of $ {\mathcal U}_0^n $ depicted as the curvy blue line. It endows $ {\mathcal U}_0^n $ of a fiber bundle structure of base $ {\mathcal N} $. Let $ \Sigma \in {\mathcal U}_0^n $. It projects (in the bundle sense) onto $ A_\Omega \in {\mathcal N} $ and so, belongs to the fiber $ {\mathcal F}_\Omega $ represented by the curvy red line. The tangent space to $ {\mathcal N} $ at $ A_\Omega $, $ T_{A_\Omega} {\mathcal N} $ is represented by the magenta straight line. Its orthogonal $ (T_{A_\Omega} {\mathcal N})^\bot $ is the gray-shaded plane on the figure. It contains $ {\mathcal F}_\Omega $ by virtue of Lemma 5.6 (ii). Then, condition (101) means that the GCI associated with $ (\eta,\Sigma) $ are the functions $ \psi $ that cancel $ L_{\eta \Sigma} f $ for all $ f $ whose Q-tensor $ Q_f $ (represented by the point Q on the figure) belongs to $ (T_{A_\Omega} {\mathcal N})^\bot $

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