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 $G:={ℝ}^{n-1}\times ℝ/L\mathbb{Z}$ to obtain an $ℛ$-bound for the...

We present a simple mechanism of cell motility in a confined geometry, inspired by recent motility assays in microfabricated channels. This mechanism relies mainly on the coupling of actin polymerisation at the cell membrane to geometric confinement. We first show analytically using a minimal model of polymerising viscoelastic gel confined in a narrow channel that spontaneous motion occurs due to polymerisation alone. Interestingly, this mechanism...

The numerical solution of the flow of a liquid crystal governed by a particular instance of the Ericksen–Leslie equations is considered. Convergence results for this system rely crucially upon energy estimates which involve $H^2\left(\Omega \right)$ norms of the director field. We show how a mixed method may be used to eliminate the need for Hermite finite elements and establish convergence of the method.

The numerical solution of the flow of a liquid crystal governed by a particular instance of the Ericksen–Leslie equations is considered. Convergence results for this system rely crucially upon energy estimates which involve H2(Ω) norms of the director field. We show how a mixed method may be used to eliminate the need for Hermite finite elements and establish convergence of the method.