On the existence of distributional potentials

Description

We present proofs for the existence of distributional potentials 𝐹 ∈ ′(Ω) for distributional vector fields𝐺 ∈ ′(Ω)𝑛, that is, grad 𝐹 = 𝐺, where Ω is an open subset of ℝ𝑛. The hypothesis in these proofs is the compatibility condition 𝜕𝑗𝐺𝑘 = 𝜕𝑘𝐺𝑗 for all 𝑗, 𝑘 ∈ {1, … , 𝑛}, if Ω is simply connected, and a stronger condition in the general case. A key tool in our treatment is the Bogovski˘ı formula, assigning vector fields 𝑣 ∈ (Ω)𝑛 satisfying div 𝑣 = 𝜑 to functions 𝜑 ∈ (Ω) with ∫ 𝜑(𝑥) d𝑥 = 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.

Links & Downloads

1. urn:nbn:de:bsz:14-qucosa2-901167
2. https://tud.qucosa.de/id/qucosa%3A90116
3. https://tud.qucosa.de/api/qucosa%3A90116/attachment/ATT-0/

Tags

info:eu-repo/classification/ddc/510

ddc:510

Additional Fields

Identifer	oai:union.ndltd.org:DRESDEN/oai:qucosa:de:qucosa:90116
Date	19 April 2024
Creators	Voigt, Jürgen
Publisher	Wiley-VCH
Source Sets	Hochschulschriftenserver (HSSS) der SLUB Dresden
Language	English
Detected Language	English
Type	info:eu-repo/semantics/publishedVersion, doc-type:article, info:eu-repo/semantics/article, doc-type:Text
Rights	info:eu-repo/semantics/openAccess
Relation	1522-2616, 10.1002/mana.202100220