Romen Semenovich Saks

arXiv2022

Curl and gradient of divergence operators in Spaces $ \mathbf{W}^{m}$ and $\mathbf{A}^{2k}$ vortex and potential fields and in the classes $\mathbf{C}(2k, m)$

The properties of the curl 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 classes $ \mathbf{C}(2k, m)(G)\equiv \mathbf{A}^{2k}(G) \oplus \mathbf{W}^m(G)$. The space $ \mathbf {L}_{2} (G) $ is decomposed into orthogonal subspaces $ \mathcal{A} $ and $ \mathcal {B} $: $\mathbf{L}_{2}(G)=\mathcal{A}\oplus \mathcal{B}$. In turn, $ \mathcal{A}= \mathcal{A}_H\oplus \mathbf{A}^0$ and $\mathcal{B}=\mathcal{B}_H \oplus \mathbf{V}^0$, where $\mathcal{A}_H $ and $\mathcal{B}_H $ are null spaces of operators $\nabla \text{div}$ and $ \text{rot}$ in $\mathcal{A}$ and $\mathcal{B}$; the dimensions of $\mathcal{A}_H $ and $\mathcal{B}_H $ are finite and determined by the topology of the boundary; $\mathcal{A}_H=\emptyset $ and $\mathcal{B}_H= \emptyset $ if the domain $Ω$ is a ball. The orthonormal basis are constructed in the class $ \mathbf{A}^0$ (resp., In $\mathbf{V}^0$ ) by eigenfields $\mathbf{q}_{j}(\mathbf{x})$ of $\nabla \text{div}$ operator (resp., $\mathbf{q}^{\pm }_{j}(\mathbf{x})$ of $ \text{rot}$ operator) with nonzero eigenvalues $μ_{j}$ (resp., $\pm λ_{j}$ ). The operators $\nabla\mathrm{div}$ and $\mathrm{rot}$ cancel each other out and project $\mathbf{L}_{2}(G) $ onto $ \mathcal {A} $ and $ \mathcal { B} $, and $ \mathrm {rot} \, \mathbf {u} = 0 $ for $ \mathbf {u} \in \mathcal {A} $, and $ \nabla \mathrm div \mathbf {v} = 0 $ for $ \mathbf {v} \in \mathcal {B} $ \cite{hw}. Laplace matrix operator expressed through them: $\mathrmΔ \mathbf {v} \equiv \nabla \mathrm{div}\,\mathbf {v} -(\mathrm{rot})^2\, \mathbf {v}$.

math.AP↗

arXiv2021

Vortex and the Gradient of Divergence in Sobolev Spaces

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.

math.AP↗