Vortex and the Gradient of Divergence in Sobolev Spaces

TL;DR

This paper investigates the properties of vortex and divergence gradient operators within Sobolev spaces on bounded domains, extending classical boundary value problem theory to new functional frameworks.

Contribution

It introduces and analyzes Sobolev space analogs for vortex and divergence operators, expanding the functional analytic foundation for boundary value problems involving these operators.

Findings

Defined new Sobolev space classes for vortex and divergence operators.

Established properties and relationships of these operators in the new spaces.

Open problems for studying powers of these operators in Sobolev spaces.

Abstract

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.