On the existence of distributional potentials

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.