A heat flow problem from Ericksen's model for nematic liquid crystals with variable degree of orientation. II (Q6136045)

The author investigates the system \begin{align*} s_t&= \Delta s-\frac{|\nabla u|^2-|\nabla s|^2}{2s^2}s-\frac{W^{\prime}(s)}{s}s,\\ u_t&= \Delta u+\frac{|\nabla u|^2-|\nabla s|^2}{2s^2}u-\frac{W^{\prime}(s)}{s}u, \end{align*} where \(W\) is assumed to be a double-well potential function. The author gives conditions under which the considered system has a \(H^1\)-solution \((s, u): \Omega\to \mathcal{C}\) that satisfies the initial-boundary conditions \begin{align*} (s(x, 0), u(x, 0))&= (g(x), h(x)),\quad x\in \Omega,\\ (s(x, t), u(x, t))&= (g(x), h(x)),\quad x\in \partial \Omega,\quad t\geq0. \end{align*} Here \(\Omega\) is a bounded domain in \(\mathbb{R}^m\) with smooth boundary, \[ \mathcal{C}=\{(s, u)\in \mathbb{R}\times \mathbb{R}^3: s^2=|u|^2\}, \] and \((g, h)\) is a Lipschitz map from \(\Omega\) into \(\mathcal{C}\) and \(-\frac{1}{2}<g(x)<1\), \(x\in \Omega\). In the paper are given conditions under which the solutions of the investigated system is Hölder continuous on \(\Omega\times (0, \infty)\). For Part I, see [the author, Commun. Anal. Geom. 26, No. 2, 411--433 (2018; Zbl 1393.76011)].