# A right inverse of curl operator which is divergence free invariant and some applications to generalized Vekua type problems

## TL;DR

This paper develops a divergence-free, curl-invariant right inverse operator for vector fields in bounded domains, with applications to Beltrami fields, Vekua problems, and Maxwell's equations, advancing mathematical tools for vector calculus and PDEs.

## Contribution

It introduces a novel divergence-free, curl-invariant right inverse of the curl operator and applies it to solve various PDE problems in physics and mathematics.

## Key findings

- Constructed a bounded right inverse of curl operator.
- Extended the inverse to divergence-free fields with boundary conditions.
- Applied the inverse to problems in electromagnetism and fluid dynamics.

## Abstract

In this work, we investigate the system formed by the equations $\text{div } \vec w=g_0$ and $\text{curl } \vec w=\vec g$ in bounded star-shaped domains of $\mathbb{R}^3$. A Helmholtz-type decomposition theorem is established based on a general solution of the above-mentioned div-curl system which was previously derived in the literature. When $g_0\equiv 0$, we readily obtain a bounded right inverse of $\text{curl}$ which is a divergence-free invariant. The restriction of this operator to the subspace of divergence-free vector fields with vanishing normal trace is the well-known Biot--Savart operator. In turn, this right inverse of $\text{curl}$ will be modified to guarantee its compactness and satisfy suitable boundary-value problems. Applications to Beltrami fields, Vekua-type problems as well as Maxwell's equations in inhomogeneous media are included.