SIAM Journal on Mathematical Analysis, Volume 52, Issue 3, Page 2201-2236, January 2020.
In order to study the asymptotic behavior of a fluid in a domain of small thickness $\varepsilon$, it is common to use that the norm of the pressure $p_\varepsilon$ in $L^q$, $q>1$, is smaller than $C\|\nabla p_\varepsilon\|_{W^{-1,q}}/\varepsilon$. Our purpose in the present paper is to improve this estimate by showing that in fact $p_\varepsilon$ can be decomposed as the sum of two terms: the first one is of order $1/\varepsilon$ with respect to $\nabla p_\varepsilon$ but it belongs to the Sobolev space $W^{1,q}$ and not only to $L^q$; the second one only belongs to $L^q$ but it is of order one with respect to $\nabla p_\varepsilon$. This result also allows us to improve the classical estimate for Korn's constant in an elastic thin domain providing a decomposition of the deformation which contains terms with a stronger regularity. The advantage of these expansions is that they simplify the study of the asymptotic behavior of continuum mechanics problems in thin domains since they give an additional compactness. As examples we provide two applications in fluid mechanics and linear elasticity.

中文翻译：

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薄域中流体压力的分解结果和弹性问题的扩展

SIAM数学分析杂志，第52卷，第3期，第2201-2236页，2020年1月。
为了研究流体在较小厚度$ \ varepsilon $的区域中的渐近行为，通常使用$ L ^ q $，$ q> 1 $，小于$ C \ | \ nabla p_ \ varepsilon \ | __W {{-1，q}} / \ varepsilon $。我们在本文中的目的是通过证明实际上$ p_ \ varepsilon $可以分解为两个项的总和来改进此估计：第一个相对于$ \ nabla p_ \具有阶数$ 1 / \ varepsilon $。 varepsilon $，但它属于Sobolev空间$ W ^ {1，q} $，不仅属于$ L ^ q $；第二个仅属于$ L ^ q $，但是相对于$ \ nabla p_ \ varepsilon $，它是一阶的。此结果还使我们能够改进弹性薄域中Korn常数的经典估计，从而提供包含具有更强规则性的项的变形分解。这些扩展的优点是，由于它们提供了额外的紧凑性，因此它们简化了薄域中连续力学问题的渐近行为的研究。作为示例，我们提供了流体力学和线性弹性的两种应用。