The current paper is devoted to the investigation of the global-in-time stability of large solutions to the compressible liquid crystal equations in the whole space. Suppose that the density is bounded from above uniformly in time in the Höder space $C^\alpha$ with $\alpha$ sufficiently small and in $L^\infty$ space respectively. Then we prove two results: (1) Such kind of the solution will converge to its associated equilibrium with a rate which is the same as that for the heat equation. (2) Such kind of the solution is stable, which means any perturbed solution will remain close to the reference solution if initially they are close to each other. This implies that the set of the smooth and bounded solutions is open.

中文翻译：

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液晶3-D可压缩流的大型解决方案的全局稳定性

本文致力于研究整个空间中可压缩液晶方程的大解的全局时间稳定性。假设密度在时间上均匀地在Höder空间$ C ^ \ alpha $中均匀分布，其中$ \ alpha $足够小，在$ L ^ \ infty $空间中。然后我们证明两个结果：（1）这种解将以与热方程相同的速率收敛到其关联的平衡。（2）这种溶液是稳定的，这意味着任何扰动的溶液如果最初彼此靠近，都将保持与参考溶液接近。这意味着光滑解和有界解的集合是开放的。