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We consider the dynamics of spatially periodic nematic liquid crystal flows in the whole space and prove existence and uniqueness of local-in-time strong solutions using maximal Lp-regularity of the periodic Laplace and Stokes operators and a local-intime existence theorem for quasilinear parabolic equations à la Clément-Li (1993). Maximal regularity of the Laplace and the Stokes operator is obtained using an extrapolation theorem on the locally compact abelian group $G: = \mathbb{R}^{n - 1} \times \mathbb{R}/L\mathbb{Z}$ to obtain an R-bound for the resolvent estimate. Then, Weis’ theorem connecting R-boundedness of the resolvent with maximal Lp regularity of a sectorial operator applies.

Maximal regularity of the spatially periodic stokes operator and application to nematic liquid crystal flows