Gevrey analyticity of solutions to the 3D nematic liquid crystal flows in critical Besov space

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• 作者：Qiao Liu ; ^{liuqao2005@163.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Nematic liquid crystal flows ; Energy argument ; Besov space ; Gevrey analyticity
• 刊名：Nonlinear Analysis: Real World Applications
• 出版年：2016
• 出版时间：October 2016
• 年：2016
• 卷：31
• 期：Complete
• 页码：431-451
• 全文大小：724 K

文摘

We show that the solution to the Cauchy problem of the 3D nematic liquid crystal flows, with initial data belonging to a critical Besov space, belongs to a Gevrey class. More precisely, it is proved that for any $(u_0,d_0-{\overline{d}}_0)\in {\dot{B}}_{p,1}^{\frac{3}{p}-1}\left({\mathbb{R}}^3\right)\times {\dot{B}}_{q,1}^{\frac{3}{q}}\left({\mathbb{R}}^3\right)$ with some suitable conditions imposed on p,q∈(1,∞)$p,q\in (1,\infty )$, there exists T^∗>0$T^{\ast }>0$ depending only on initial data, such that the nematic liquid crystal flows admit a unique solution (u,d)$(u,d)$ on R³×(0,T^∗)${\mathbb{R}}^3\times (0,T^{\ast })$, and satisfies Here, ${\overline{d}}_0\in {\mathbb{S}}^2$ is a constant unit vector, and Λ₁${\Lambda }_1$ is the Fourier multiplier whose symbol is given by |ξ|₁=|ξ₁|+|ξ₂|+|ξ₃|${\left|\xi \right|}_1=\left|{\xi }_1\right|+\left|{\xi }_2\right|+\left|{\xi }_3\right|$. Moreover, if the initial data is sufficiently small, then T^∗=∞$T^{\ast }=\infty$. As a consequence of the results, decay estimates of higher-order derivatives of solutions in Besov spaces are deduced.