A heat flow problem from Ericksen’s model for nematic liquid crystals with variable degree of orientation, II

We study a heat flow problem for nematic liquid crystals with variable degree of orientation. Let $\Omega$ be a bounded domain in $\mathbb{R}^m$ with smooth boundary and $\mathcal{C}$ be the round cone in $\mathbb{R}^ \times \mathbb{R}^3$,\[\mathcal{C} = {\lbrace (s, u) \in \mathbb{R}^ \times \mathbb{R}^3 \; : \quad s^2 = {\lvert u \rvert}^2 \rbrace} \textrm{.}\]Under certain conditions on the double-well potential function$W(s)$, we prove that there exist solutions $(s, u) : \Omega \times [0,\infty) \to \mathcal{C}$which satisfy the system\begin{align}s^t = \Delta s - {\lvert \nabla u\rvert}^2 - \frac{{\lvert \nabla s \rvert}^2}{2s^2} s - \frac{W^\prime (s)}{s} s \\u^t = \Delta u - {\lvert \nabla u\rvert}^2 - \frac{{\lvert \nabla s \rvert}^2}{2s^2} u - \frac{W^\prime (s)}{s} u\end{align}with given initial-boundary data. Also, we prove that the solutions are Holder continuous.

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The author’s research is supported in part by the NSC (Taiwan).grant 108-2115-M-194-003.

Received 27 June 2019

Accepted 18 May 2020

Published 17 August 2023