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 $\Gamma$ 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 $\Delta^m\,u=\rho$ in the spaces $W_2^m(\Omega)$ 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.