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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.

76A15 Liquid crystals
35Q35 PDEs in connection with fluid mechanics
Full Text: DOI
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