On the existence of distributional potentials

We present proofs for the existence of distributional potentials F∈D′(Ω)$F\in \mathcal {D}^{\prime }(\Omega )$ for distributional vector fields G∈D′(Ω)n$G\in \mathcal {D}^{\prime }(\Omega )^n$, that is, gradF=G$\operatorname{grad}F=G$, where Ω is an open subset of Rn$\mathbb {R}\nonscript \hspace{0.29999pt}^n$. The hypothesis in these proofs is the compatibility condition ∂jGk=∂kGj$\partial _jG_k=\partial _kG_j$ for all j,k∈{1,⋯,n}$j,k\in \lbrace 1,\dots ,n\rbrace$, if Ω is simply connected, and a stronger condition in the general case. A key tool in our treatment is the Bogovskiĭ formula, assigning vector fields v∈D(Ω)n$v\in \mathcal {D}(\Omega )^n$ satisfying divv=φ$\operatorname{div}v=\varphi$ to functions φ∈D(Ω)$\varphi \in \mathcal {D}(\Omega )$ with ∫φ(x)dx=0$\int \hspace{-3.05542pt}\varphi (x)\mathclose {}\,\mathrm{d}x=0$. The results are applied to properties of Hilbert spaces of functions occurring in the treatment of the Stokes operator and the Navier–Stokes equations.