On the existence of distributional potentials

Abstract

We present proofs for the existence of distributional potentials $F\in {\mathcal{D}}^{\prime }(\Omega )$ for distributional vector fields $G\in {\mathcal{D}}^{\prime }(\Omega {)}^n$, i.e. $\mathrm{grad}F=G$, where $\Omega$ is an open subset of ${\mathbb{R}}^n$. The hypothesis in these proofs is the compatibility condition ${\partial }_j{G}_k={\partial }_k{G}_j$ 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 $\mathrm{div}v=\phi$ to functions $\phi \in \mathcal{D}(\Omega )$ with $\int \phi (x)\mathrm{d}x=0$. The results are applied to properties of Hilbert spaces of functions occurring in the treatment of the Stokes operator and the Navier--Stokes equations.