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Czechoslovak Mathematical Journal, Vol. 66, No. 1, pp. 41-55, 2016
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Maximal regularity of the spatially periodic Stokes operator and application to nematic liquid crystal flows

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Jonas Sauer

**Abstract:** 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 $L^p$-regularity of the periodic Laplace and Stokes operators and a local-in-time existence theorem for quasilinear parabolic equations à la ClĂ©ment-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 $\mathcal{R}$-bound for the resolvent estimate. Then, Weis' theorem connecting $\mathcal{R}$-boundedness of the resolvent with maximal $L^p$ regularity of a sectorial operator applies.

**Keywords:** Stokes operator; spatially periodic problem; maximal $L^p$ regularity; nematic liquid crystal flow; quasilinear parabolic equations

**Classification (MSC 2010):** 35B10, 35K59, 35Q35, 76A15, 76D03

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