论文标题
周期性设置中的近晶液晶模型
A smectic liquid crystal model in the periodic setting
论文作者
论文摘要
我们将渐近行为视为$ \ varepsilon $在\ begin {equation*} \ mathcal {e} _ {\ varepsilon}（w）给定的周期性设置中，将2D晶格模型的零变为零。 = \ frac {1} {2} \ int _ {\ mathbb {t} ^{2}}} \ frac {1} {\ varepsilon} \ left（\ left \ left \ vert \ vert \ partial_ {1} \ partial_ {2} w- \ partial_ {1} \ frac {1} {2} w ^{2} \ right）\ right） ^{2}+\ varepsilon \ left（\ partial_ {1} w \ right） \ end {equation*}我们表明，能量$ \ Mathcal {e} _ \ varepsilon（w）$控制合适的$ l^p $和$ w $的besov规范，并用它来证明对$ \ \ \ \ \ m rathcal {e} _ \ varepsilon（w varepsilon（w varepsilon）的最小化的存在，以前曾被证明过了这一模型$ l^p $用于限制能量的序列。我们还通过熵参数证明了$ \ mathcal {e} _ \ varepsilon（w）$的渐近下限为$ \ varepsilon \ to 0 $。
We consider the asymptotic behavior as $\varepsilon $ 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 $\mathcal{E}_\varepsilon(w)$ controls suitable $L^p$ and Besov norms of $w$ and use this to demonstrate the existence of minimizers for $\mathcal{E}_\varepsilon(w)$, 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 $\mathcal{E}_\varepsilon(w)$ as $\varepsilon \to 0$ by means of an entropy argument.
