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Keywords:
Stokes operator; spatially periodic problem; maximal $L^p$ regularity; nematic liquid crystal flow; quasilinear parabolic equations
Stokes operator; spatially periodic problem; maximal $L^p$ regularity; nematic liquid crystal flow; quasilinear parabolic equations
Summary:
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 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 $\mathcal {R}$-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.
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