On the Multi-Dimensional Divergence-Curl Problem and Its Connection with Pseudo-Harmonic Fields

TL;DR

This paper investigates the conditions for solving the multi-dimensional divergence-curl problem with boundary constraints, introducing a criterion based on orthogonality to pseudo-harmonic fields and providing a specific family of such fields for 3D exterior problems.

Contribution

It establishes a new solvability criterion involving pseudo-harmonic fields and presents a countable family of these fields for 3D exterior problems.

Findings

Derived a solvability criterion as an orthogonality condition.

Presented a countable family of pseudo-harmonic fields for 3D exterior problems.

Abstract

This article addresses the solvability of the multi-dimensional divergence-curl problem with a no-slip boundary condition. A solvability criterion is derived as an orthogonality condition of the vorticity function to pseudo-harmonic fields. A countable family of such fields, sufficient for the solvability of the three-dimensional problem in the exterior of a sphere, is also presented.