A smectic liquid crystal model in the periodic setting

Abstract

We consider the asymptotic behavior as $\epsilon$ goes to zero of the 2D smectics model in the periodic setting given by \begin{equation*} \mathcal{E}_{\varepsilon }( w) =\frac{1}{2}\int_{\mathbb{T}^{2}}\frac{1}{ \varepsilon }\left( \left\vert \partial_{1}\right\vert ^{-1}\left( \partial_{2}w-\partial_{1}\frac{1}{2}w^{2}\right) \right) ^{2}+\varepsilon \left( \partial_{1}w\right) ^{2}dx . \end{equation*} We show that the energy ${\mathcal{E}}_{\epsilon }(w)$ controls suitable $L^p$ and Besov norms of $w$ and use this to demonstrate the existence of minimizers for ${\mathcal{E}}_{\epsilon }(w)$, which has not been proved for this smectics model before, and compactness in $L^p$ for an energy-bounded sequence. We also prove an asymptotic lower bound for ${\mathcal{E}}_{\epsilon }(w)$ as $\epsilon \to 0$ by means of an entropy argument.

The published version in Nonlinear Analysis is available at http://dx.doi.org/https://doi.org/10.1016/j.na.2022.113187