Network of Sobolev spaces and boundary value problems for operators vortex and gradient of divergence

We will consider the scale of the Sobolev spaces ${\mathbf{\text{H}}}^m\left(G\right)$ vector fields in a bounded domain G of ${\mathrm{ℝ}}^3$ with a smooth boundary of $\Gamma$. The gradient-ofdivergence and the rotor-of-rotor operators ($\nabla \text{div}$ and ${\text{rot}}^2$) and their powers are analogous to the scalar operator ${\Delta }^m$ in ${\mathrm{ℝ}}^3$. They generate spaces ${\mathbf{\text{A}}}^{\text{2}\mathrm{k}}\text{(}G\text{)}$ and ${\mathbf{\text{W}}}^m\left(G\right)$ potential and vortex fields; where the numbers k, m > 0 are integers.
It is proven that ${\mathbf{\text{A}}}^{\text{2}\mathrm{k}}\text{(}G\text{)}$ and ${\mathbf{\text{W}}}^m\left(G\right)$ are projections of Sobolev spaces ${\mathbf{\text{H}}}^{2k}\left(G\right)$ and ${\mathbf{\text{H}}}^m\left(G\right)$ in subspaces $\mathcal{A}$ and $\mathcal{B}$ in ${\mathbf{\text{L}}}_2\left(G\right)$. Their direct sums ${\mathbf{\text{A}}}^{2k}\left(G\right)\oplus {\mathbf{\text{W}}}^m\left(G\right)$ form a network of spaces. Its elements are classes $\mathbf{\text{C}}\text{(}2k,m\text{)≡}{\mathbf{\text{A}}}^{2k}\oplus {\mathbf{\text{W}}}^m$.
We consider at the properties of the spaces ${\mathbf{\text{A}}}^{-m}$ and ${\mathbf{\text{W}}}^{-m}$ and proved their compliance with the spaces ${\mathbf{\text{A}}}^m$ and ${\mathbf{\text{W}}}^m$. We also consider at the direct sums of ${\mathbf{\text{A}}}^k\left(G\right)\oplus {\mathbf{\text{W}}}^m\left(G\right)$ for any integer numbers k and m>0. This completes the construction of the ${\left\{\mathbf{\text{C}}(k,m)\right\}}_{k,m}$ network.
In addition, an orthonormal basis has been constructed in the space ${\mathbf{\text{L}}}_2\left(G\right)$. It consists of the orthogonal subspace $\mathcal{A}$ and $\mathcal{B}$ bases. Its elements are eigenfields of the operators $\nabla \text{div}$ and $\text{rot}$. The proof of their smoothness is an important stage in the theory developed.
The model boundary value problems for the operators $\text{rot}+\lambda I$, $\nabla \text{div}+\lambda I$, their sum, and also for the Stokes operator have been investigated in the network ${\left\{\mathbf{\text{C}}(k,m)\right\}}_{k,m}$. Solvability conditions are obtained for the model problems considered.