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On well-posedness and large time behavior for smectic-A liquid crystals equations in $$\mathbb {R}^3$$ R 3
Zeitschrift für angewandte Mathematik und Physik ( IF 2 ) Pub Date : 2020-10-07 , DOI: 10.1007/s00033-020-01407-4
Xiaopeng Zhao , Yong Zhou

The main purpose of this manuscript is to study the well-posedness and decay estimates for strong solutions to the Cauchy problem of 3D smectic-A liquid crystals equations. First, applying Banach fixed point theorem, we prove the local existence and uniqueness of strong solutions. Then, by establishing some nontrivial estimates with energy method and a standard continuity argument, we prove that there exists a unique global strong solution provided that the initial data are sufficiently small. Moreover, we also establish the suitable negative Sobolev norm estimates and obtain the optimal decay rates of the higher-order spatial derivatives of the strong solutions.



中文翻译:

$$ \ mathbb {R} ^ 3 $$ R 3中近晶A液晶方程的适定性和长时间行为

该手稿的主要目的是研究3D层列A液晶方程的柯西问题强解的适定性和衰减估计。首先,应用Banach不动点定理,证明强解的局部存在性和唯一性。然后,通过使用能量方法和标准连续性参数建立一些非平凡的估计,我们证明只要初始数据足够小,就存在唯一的全局强解。此外,我们还建立了合适的负Sobolev范数估计,并获得了强解的高阶空间导数的最佳衰减率。

更新日期:2020-10-07
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