Global Well-Posedness and Temporal Decay Estimates for the 3D Nematic Liquid Crystal Flows

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Abstract

In this paper, we investigate global well-posedness and large time behavior of the Cauchy problem for the 3D incompressible nematic liquid crystal flows. By using the advantage of suitable weighted function, we show that for any initial data \((u_{0},d_{0}-\overline{d}_{0})\) in critical Besov spaces \(\dot{B}^{\frac{3}{p}-1}_{p,1}(\mathbb {R}^{3})\times \dot{B}^{\frac{3}{q}}_{q,1}(\mathbb {R}^{3})\) with \(1< p, q<\infty \) and \( -\inf \{\frac{1}{3},\frac{1}{2p}\}\le \frac{1}{q}-\frac{1}{p}\le \frac{1}{3}\), if the initial orientation \(d_{0}-\overline{d}_{0}\) and a certain nonlinear function of initial velocity \(u_{0}\) are small enough, then there exists a global-in-time solution to the nematic liquid crystal flows. We also give an example of initial velocity satisfying that nonlinear smallness condition, but each component of its norm may be arbitrarily large. Moreover, if we further assume that \((u_{0},d_{0}-\overline{d}_{0})\in \dot{B}^{-s}_{r,\infty }(\mathbb {R}^{3})\times \dot{B}^{-s+1}_{r,\infty }(\mathbb {R}^{3})\) with \(1<r\le \inf \{p,q\}\) and \( \sup \{0,1-\frac{3}{r}\}\le s<\inf \{4-\frac{3}{r},1+\frac{3}{p},1+\frac{3}{q}\}\), then the global-in-time strong solution (ud) to the nematic liquid crystal flows admits the following temporal decay rate
$$\begin{aligned}&\Vert {u}(t)\Vert _{\dot{B}^{\ell }_{p,1}} \le C(1+t)^{-\frac{\ell +s}{2}-\frac{3}{2}\left( \frac{1}{r}-\frac{1}{p}\right) } \quad \text { for all } -s-3\left( \frac{1}{r}-\frac{1}{p}\right)<\ell \le \frac{3}{p}-1;\\&\Vert {d}(t)-\overline{d}_{0}\Vert _{\dot{B}^{\ell }_{q,1}} \le C(1+t)^{-\frac{\ell +s+1}{2}-\frac{3}{2}\left( \frac{1}{r}-\frac{1}{q}\right) } \quad \text { for all } -s+1-3\left( \frac{1}{r}-\frac{1}{q}\right) <\ell \le \frac{3}{q}. \end{aligned}$$
Here, \(\overline{{d}}_{0}\in \mathbb {S}^{2}\) is a constant unit vector.

Keywords

Nematic liquid crystal flows Global well-posedness Temporal decay estimate Besov space 

Mathematics Subject Classification

76A15 35B65 35Q35 

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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Key Laboratory of High Performance Computing and Stochastic Information Processing (HPCSIP) (Ministry of Education of China)College of Mathematics and Computer Science, Hunan Normal UniversityChangshaPeople’s Republic of China

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