Finite Dimensional Reduction and Convergence to Equilibrium for Incompressible Smectic-A Liquid Crystal Flows
Finite Dimensional Reduction and Convergence to Equilibrium for Incompressible Smectic-A Liquid Crystal Flows
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DOI：
10.1137/100813427
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发表时间：
2010-11
期刊：
影响因子：
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通讯作者：
A. Segatti;Hao Wu
中科院分区：
文献类型：
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作者：
A. Segatti;Hao Wu
我们考虑了一个流体动力系统，该系统模拟了smetic - a液晶流动。该模型由流体速度的Navier-Stokes方程和层变量的四阶方程组成，并赋予周期边界条件。我们分析了无限维耗散动力系统理论中解的长时间行为。首先证明了在二维空间中，该问题在一定相空间中具有一个全局吸引子$\mathcal{a}$。然后我们建立了指数吸引子$\mathcal{M}$的存在性，这意味着全局吸引子$\mathcal{A}$具有有限的分形维数。此外，我们通过一个合适的Łojasiewicz-Simon不等式证明了每个轨迹收敛到一个单一的平衡。并讨论了三维空间的相应结果。
We consider a hydrodynamic system that models smectic-A liquid crystal flow. The model consists of the Navier–Stokes equation for the fluid velocity coupled with a fourth-order equation for the layer variable, endowed with periodic boundary conditions. We analyze the long-time behavior of the solutions within the theory of infinite-dimensional dissipative dynamical systems. We first prove that in two dimensions, the problem possesses a global attractor $\mathcal{A}$ in a certain phase space. Then we establish the existence of an exponential attractor $\mathcal{M}$, which entails that the global attractor $\mathcal{A}$ has finite fractal dimension. Moreover, we show that each trajectory converges to a single equilibrium by means of a suitable Łojasiewicz–Simon inequality. Corresponding results in three dimensions are also discussed.