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arXiv 2201.08818math.AP
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索伯列夫空间中的涡与散度梯度

Vortex and the Gradient of Divergence in Sobolev Spaces

Romen Semenovich Saks

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in Russian
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摘要：

研究了$ \mathbf {L}_2 (G) $空间、具有光滑边界的有界区域$ G \subset \textrm {R}^3 $和索伯列夫空间：$ \mathbf{C}(2k, m)(G)\equiv \mathbf{A}^{2k}(G) \oplus \mathbf{W}^m(G)$中的涡旋算子和散度梯度算子（$ \text{rot}$和$\nabla \text{div}$）的性质. S.L. Sobolev研究了标量多调和方程$Δ^m\,u=ρ$在空间$W_2^m(Ω)$广义右手边的边值问题，奠定了这些空间的理论基础。它的结构有矩阵类似物，这里是其中的一些。类$ \mathcal {A} $和$ \mathcal {B} $中的空间${W}_2^{(m)}(G)$的类似物是$ 2k> 0 $和$ m> 0 $阶的空间$\mathbf{A}^{2k}(G)$和$\mathbf{W}^m(G)$，以及$ \mathbf {A}^{-2k} (G) $和它们的对偶空间$ \mathbf{W}^{- m}(G) $。空间对形成索伯列夫空间的网，其元素是类$ \mathbf{C}(2k, m)(G)\equiv \mathbf{A}^{2k}(G) \oplus \mathbf{W}^m(G)$ ;类$ \mathbf{C}(2k, 2k)$与索伯列夫空间$\mathbf{H}^{2k}(G)$重合。它们属于$\mathbf{L}_{2}(G)$，如果是$k\geq 0$和$m\geq 0$。一个广泛的问题领域已经打开：研究运营商$(\mathrm{rot})^p$，$ (\nabla \, \mathrm{div})^p$的$ p = 1,2, ...,$和其他网络索伯列夫空间。

英文摘要：

The properties of the vortex and the gradient of divergence operators ( $ \text{rot}$ and $\nabla \text{div}$ ) are studied in the space $ \mathbf {L}_2 (G) $ in a bounded domain $ G \subset \textrm {R}^3 $ with a smooth boundary $ Γ$ and in the Sobolev spaces: $ \mathbf{C}(2k, m)(G)\equiv \mathbf{A}^{2k}(G) \oplus \mathbf{W}^m(G)$. S.L. Sobolev studied boundary value problems for the scalar polyharmonic equation $Δ^m\,u=ρ$ in the spaces $W_2^m(Ω)$ with a generalized right-hand side and laid the foundation for the theory of these spaces. Its constructions have matrix analogs, here are some of them. Analogues of the spaces ${W}_2^{(m)}(G)$ in the classes $ \mathcal {A} $ and $ \mathcal {B} $ are the space $\mathbf{A}^{2k}(G)$ and $\mathbf{W}^m(G)$ of orders $ 2k> 0 $ and $ m> 0 $, and $ \mathbf {A}^{-2k} (G) $ and their dual spaces $ \mathbf{W}^{- m}(G) $. Pairs of spaces form a net of Sobolev spaces, its elements are classes $ \mathbf{C}(2k, m)(G)\equiv \mathbf{A}^{2k}(G) \oplus \mathbf{W}^m(G)$; the class $ \mathbf{C}(2k, 2k)$coincides with the Sobolev space $\mathbf{H}^{2k}(G)$. They belong to $\mathbf{L}_{2}(G)$, if $k\geq 0$ and $m\geq 0$. A wide field of problems has opened up: studying the operators $(\mathrm{rot})^p$, $ (\nabla \, \mathrm{div})^p$ for $ p = 1,2, ...,$ and others in the network Sobolev spaces.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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