On a double penalized smectic-A model

Abstract

In smectic-A liquid crystals a unity director vector n appear, modeling an average preferential direction of the molecules and also the normal vector of the layer configuration. In the E’s model [5] W. E. Nonlinear Continuum Theory of Smectic-A Liquid Crystals, Arch. Rat. Mech. Anal., 137, 2 (2010), 1473-1493, the Ginzburg-Landau penalization related to the constraint |n| = 1 is considered and, assuming the constraint ∇ × n = 0, n is replaced by the so-called layer variable ϕ such that n = ∇ϕ. In this paper, a double penalized problem is introduced related to a smectic-A liquid crystal flows, considering a Cahn-Hilliard system to model the behavior of n. Then, the issue of the global in time behavior of solutions is attacked, including the proof of the convergence of the whole trajectory towards a unique equilibrium state.