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

TL;DR

This paper investigates the properties of curl and divergence gradient operators in specific function spaces within bounded domains, providing spectral decompositions and basis constructions related to vortex and potential fields.

Contribution

It introduces a detailed spectral analysis and orthogonal basis construction for curl and divergence operators in specialized function spaces, considering domain topology effects.

Findings

Decomposition of $L_2(G)$ into orthogonal subspaces $ extbf{A}$ and $ extbf{B}$.

Construction of orthonormal bases using eigenfields of curl and divergence operators.

Explicit relations between operators and the Laplace operator in these spaces.

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: $\mathrm{\Delta} \mathbf {v} \equiv \nabla \mathrm{div}\,\mathbf {v} -(\mathrm{rot})^2\, \mathbf {v}$.