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Finite dimensional reduction and convergence to equilibrium for incompressible smectic-A liquid crystal flows. (English) Zbl 1252.35079
The authors consider the following simplified hydrodynamic system with respect to a nonlinear continuum theory for smectic-A liquid crystals \begin{aligned} -{\mathbf v}_t+{\mathbf v}\cdot \nabla { \mathbf v}-\frac{\mu _4}{2}\Delta {\mathbf v} + \nabla p& = \nabla \cdot (\widetilde{\sigma } ^d + \widetilde{\sigma }^e) , \\ \nabla \cdot {\mathbf v}& = 0 , \\ \phi _t + {\mathbf v} \cdot \nabla \phi& = \lambda (-K \Delta ^2 \phi + \nabla \cdot f(\nabla \phi )),\end{aligned} where \begin{aligned} \widetilde{\sigma }^d &= \mu _1 ({\mathbf d}^T D({\mathbf v}{\mathbf d}) {\mathbf d} \otimes {\mathbf d} + \mu _5 (D({\mathbf v }) {\mathbf d } \otimes {\mathbf d} + {\mathbf d} \otimes D({\mathbf d}){\mathbf d}), \\ \widetilde{\sigma }^e &= -f({\mathbf d} )\otimes {\mathbf d} + K\nabla (\nabla \cdot {\mathbf d}) \otimes {\mathbf d} -K (\nabla \cdot {\mathbf d} )\nabla {\mathbf d}.\end{aligned} Here, $$\mu _1,\mu _5\geq 0,\;\mu_4>0$$ are dissipative coefficients in the stress tensor, $$K>0$$ is a constant, $$D({\mathbf v})$$ indicates the symmetric velocity gradient and $f({\mathbf d})= f(\nabla \phi )= \frac{1}{\epsilon ^2}(| \nabla \phi | ^2 -1) \nabla \phi ,\quad 0<\epsilon \leq 1.$ They consider the problem on the $$n$$-dimensional torus $${\mathbb T}^n= {\mathbb R}^n/{\mathbb Z}^n\;(n=2,3)$$; namely, with periodic boundary conditions.
In the three dimensional case, they show that the local existence of strong solutions for arbitrary initial data by a higher-order energy estimate. If $$\mu _4$$ is large enough, they also obtain the global existence of strong solutions. Finally, they show that the global weak/strong solutions converge to single equilibria as in the two dimensional case. In particular, they prove the well-posedness and long-time behavior of global strong solutions when the initial data are close to a local minimizer of the energy using a suitable Lojasiewics-Simon-type inequality, which improve the result of the paper C. Liu [Discrete Contin. Dyn. Syst. 6, No. 3, 591–608 (2000; Zbl 1021.35083)].

##### MSC:
 35B41 Attractors 76A15 Liquid crystals 76D05 Navier-Stokes equations for incompressible viscous fluids 35Q35 PDEs in connection with fluid mechanics
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