Long-Time Behavior for a Hydrodynamic Model on Nematic Liquid Crystal Flows with Asymptotic Stabilizing Boundary Condition and External Force
Long-Time Behavior for a Hydrodynamic Model on Nematic Liquid Crystal Flows with Asymptotic Stabilizing Boundary Condition and External Force
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DOI：
10.1137/120866476
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发表时间：
2011-11
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影响因子：
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通讯作者：
M. Grasselli;Hao Wu
中科院分区：
文献类型：
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作者：
M. Grasselli;Hao Wu
在本文中，我们考虑了一个简化的Ericksen-Leslie模型的双折射液晶流。演化系统由Navier-Stokes方程和对流Ginzburg-朗道型方程组成。我们假设Navier-Stokes方程的特征在于无滑移边界条件和与时间相关的外力$\mathbf{g}（t）$，而分子指向矢方程受与时间相关的Dirichlet边界条件$\mathbf{h}（t）$。我们证明了在二维情形下，当$\mathbf{h}（t）$和$\mathbf{g}（t）$分别收敛到与时间无关的边界点$\mathbf{h}_\infty$和$\mathbf{0}$时，每个整体弱解收敛到一个定态.最后给出了收敛速度的估计。在三维的情况下，我们证明了整体弱解最终是强的，这样就可以证明类似于二维情况的结果。我们还证明了存在的globa...
In this paper, we consider a simplified Ericksen--Leslie model for the nematic liquid crystal flow. The evolution system consists of the Navier--Stokes equations coupled with a convective Ginzburg--Landau type equation for the averaged molecular orientation. We suppose that the Navier--Stokes equations are characterized by a no-slip boundary condition and a time-dependent external force $\mathbf{g}(t)$, while the equation for the molecular director is subject to a time-dependent Dirichlet boundary condition $\mathbf{h}(t)$. We show that in the two-dimensional case, each global weak solution converges to a single stationary state when $\mathbf{h}(t)$ and $\mathbf{g}(t)$ converge to a time-independent boundary datum $\mathbf{h}_\infty$ and $\mathbf{0}$, respectively. Estimates on the convergence rate are also obtained. In the three-dimensional case, we prove that global weak solutions are eventually strong so that results similar to the two-dimensional case can be proved. We also show the existence of globa...