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

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 $