Vector fields with zero flux through spheres of fixed radius

The classical property of a periodic function on the real axis is the possibility of its representation by a trigonometric Fourier series. The natural analogue of the periodicity condition in the Euclidean space $\mathbb{R}^n$ is the constancy of the integrals of the function over all balls (or spheres) of a fixed radius. Functions with the specified property can be expanded in a series in special eigenfunctions of the Laplace operator. This fact admits a generalization to vector fields in $\mathbb{R}^n$, having zero flow through spheres of fixed radius. In this case, Smith's representation arises for them as the sum of a solenoidal vector field and an infinite number of potential vector fields. Potential vector fields satisfy the Helmholtz equation related to the zeros of the Bessel function $J_{n/2}$. The purpose of this paper is to obtain local analogs of the Smith theorem. We study vector fields $\mathbf{A}$ with zero flow through spheres of fixed radius on domains $\mathcal{O}$ in Euclidean space that are invariant with respect to rotations. Cases are considered when $\mathcal{O}=B_{R}=\{x\in\mathbb{R}^n: | x |<R\}$ or $\mathcal{O}=B_{a, b}= \{x\in\mathbb{R}^n: a <| x | <b\} $. The description of the fields $\mathbf{A}$ consists of two steps. The first step proves the equality $\mathbf{A}({x})={\mathbf{A}}^s({x})+B({x}){x}$, ${x}\in\mathcal{O}$, where ${\mathbf{A}}^s$ is a suitable solenoidal vector field and $B$ is a scalar field. The second step is to describe the functions $B(x)$. As the main tool for the description of $B(x)$, multidimensional Fourier series in spherical harmonics are used. If $\mathcal{O}=B_{R}$ then the Fourier coefficients of the function $B(x)$ can be represented in the form of series in the hypergeometric functions ${_1}F_2$. In the case of $\mathcal{O}=B_{a,b}$ the corresponding Fourier coefficients can be expanded in the series containing the Bessel, Neumann and Lommel functions. These results can be used in harmonic analysis of vector fields on domains in $\mathbb{R}^n$.