A smectic liquid crystal model in the periodic setting

DOI10.1016/J.NA.2022.113187arXiv2205.01872MaRDI QIDQ6398211

Xiao-Dong Yan, Michael Novack

Publication date: 3 May 2022

Abstract: We consider the asymptotic behavior as $varepsilon$ goes to zero of the 2D smectics model in the periodic setting given by �egin{equation*} mathcal{E}_{varepsilon }( w) =frac{1}{2}int_{mathbb{T}^{2}}frac{1}{ varepsilon }left( leftvert partial_{1} ightvert ^{-1}left( partial_{2}w-partial_{1}frac{1}{2}w^{2} ight) ight) ^{2}+varepsilon left( partial_{1}w ight) ^{2}dx . end{equation*} We show that the energy $mathcal{E}_{v}arepsilon(w)$ controls suitable $L^p$ and Besov norms of $w$ and use this to demonstrate the existence of minimizers for $mathcal{E}_{v}arepsilon(w)$, 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 $mathcal{E}_{v}arepsilon(w)$ as $varepsilono0$ by means of an entropy argument.

Mathematics Subject Classification ID

Statistical mechanics of liquids (82D15)

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