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We consider the asymptotic behavior as $\varepsilon $ goes to zero of the 2Dsmectics 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$\mathcal{E}_\varepsilon(w)$ controls suitable $L^p$ and Besov norms of $w$ anduse this to demonstrate the existence of minimizers for$\mathcal{E}_\varepsilon(w)$, which has not been proved for this smectics modelbefore, and compactness in $L^p$ for an energy-bounded sequence. We also provean asymptotic lower bound for $\mathcal{E}_\varepsilon(w)$ as $\varepsilon \to0$ by means of an entropy argument.

A smectic liquid crystal model in the periodic setting