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

Abstract

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 $ \Gamma$ 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 $\Omega $ 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 $\mu_{j}$ (resp., $\pm \lambda_{j}$ ). 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: $\mathrm{\Delta} \mathbf {v} \equiv \nabla \mathrm{div}\,\mathbf {v} -(\mathrm{rot})^2\, \mathbf {v}$.