On the existence of distributional potentials

TL;DR

This paper proves the existence of distributional potentials for vector fields under certain compatibility conditions, using the Bogovskii formula, with applications to the analysis of the Stokes and Navier--Stokes equations.

Contribution

It provides new proofs for the existence of distributional potentials using the Bogovskii formula, extending to non-simply connected domains and applying to fluid dynamics equations.

Findings

Existence of distributional potentials under compatibility conditions.

Use of Bogovskii formula to construct potentials.

Applications to properties of Hilbert spaces in fluid dynamics.

Abstract

We present proofs for the existence of distributional potentials $F\in{\mathcal D}'(\Omega)$ for distributional vector fields $G\in{\mathcal D}'(\Omega)^n$, i.e. $\operatorname{grad} F=G$, where $\Omega$ is an open subset of ${\mathbb R}^n$. The hypothesis in these proofs is the compatibility condition $\partial_jG_k=\partial_kG_j$ for all $j,k\in\{1,\dots,n\}$, if $\Omega$ is simply connected, and a stronger condition in the general case. A key ingredient of our treatment is the use of the Bogovskii formula, assigning vector fields $v\in{\mathcal D}(\Omega)^n$ with $\operatorname{div} v=\varphi$ to functions $\varphi\in{\mathcal D}(\Omega)$ with $\int \varphi(x)\,\mathrm{d}x=0$. The results are applied to properties of Hilbert spaces of functions occurring in the treatment of the Stokes operator and the Navier--Stokes equations.