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Global well-posedness for the Cauchy problems of the 3d simplified Ericksen-Leslie system with Fujita-Kato type initial data
  • Liu Qiao
Liu Qiao
Central South University

Corresponding Author:[email protected]

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Abstract

In this paper, we provide the global well-posedness of the Cauchy problems of the 3d simplified Ericksen-Leslie system, modelling the incompressible nematic liquid crystal flows, with Fujita-Kato type initial data. More precisely, we prove that for any initial data $(u_0,d_0)\in H^{s}(\mathbb{R}^3)\times H^{s+1}(\mathbb{R}^3)$ with $s>\frac{5}{2}$, there exists a sufficient small $\varepsilon>0$ such that if the initial velocity $u_{0}$ satisfies \begin{align*} \|u_0\|_{\dot{H}^{\frac{1}{2}}}<\varepsilon \nu, \end{align*} and the initial orientation $d_0$ satisfies \begin{align*} \|d_0\|_{\dot{H}^{\frac{1}{2}}\cap\dot{H}^{\frac{3}{2}}}^2 \exp\Big\{ C\frac{\nu+\gamma}{\nu^2\gamma}(1+\|u_0\|_{\dot{H}^\frac{1}{2}}^2)\| u_0\|_{\dot{H}^\frac{1}{2}}^2 \Big\} <\varepsilon \min\{\nu,\gamma\}, \end{align*} for some positive constant $C$, the Ericksen-Leslie system admits a unique global solution $(u,d)\in C(\mathbb{R}_+; H^{s}(\mathbb{R}^3)\times H^{s+1}(\mathbb{R}^3))$.