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-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 L^{p} regularity of a sectorial operator applies., Jonas Sauer., and Obsahuje seznam literatury
We consider the Navier-Stokes equations in unbounded domains $\Omega \subseteq \mathbb R ^n$ of uniform $C^{1,1}$-type. We construct mild solutions for initial values in certain extrapolation spaces associated to the Stokes operator on these domains. Here we rely on recent results due to Farwig, Kozono and Sohr, the fact that the Stokes operator has a bounded $H^\infty $-calculus on such domains, and use a general form of Kato's method. We also obtain information on the corresponding pressure term.