Abstract
Given a uniform $W^{3-1/q}_q$-domain $\Omega \subseteq \mathbb R^N$ for $N<q<\infty$, we consider a simplified Ericksen--Leslie system modeling the flow of compressible nematic liquid crystals based on Lin and Liu [Comm. Pure Appl. Math., 48 (1995), pp. 501--537]. We show the unique existence of local-in-time strong solutions. Furthermore, if $\Omega$ is bounded and initial data are chosen suitably small, we obtain global-in-time strong solutions. Our approach is based on maximal regularity estimates of the compressible Navier--Stokes equation by Enomoto, von Below, and Shibata [Ann. Univ. Ferrara, 60 (2014), pp. 55--89] and maximal regularity estimates for the Neumann problem as a consequence of Weis's 2001 vector-valued Fourier multiplier theorem.
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