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

Sauer, Jonas

Czechoslovak Mathematical Journal, Tome 66 (2016) no. 1, pp. 41-55.

Résumé

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.

Détail

DOI : 10.1007/s10587-016-0237-2

Classification : 35B10, 35K59, 35Q35, 76A15, 76D03
Mots-clés : Stokes operator; spatially periodic problem; maximal $L^p$ regularity; nematic liquid crystal flow; quasilinear parabolic equations