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  Helmholtz decomposition and potential functions for n-dimensional analytic vector fields

Glötzl, E., Richters, O. (2023): Helmholtz decomposition and potential functions for n-dimensional analytic vector fields. - Journal of Mathematical Analysis and Applications, 525, 2, 127138.
https://doi.org/10.1016/j.jmaa.2023.127138

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https://doi.org/10.48550/arXiv.2102.09556 (???ENUM_CONTENTCATEGORY_pre-print???)
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 ???ViewItemFull_lblCreators???:
Glötzl, Erhard1, ???ENUM_CREATORROLE_AUTHOR???
Richters, Oliver2, ???ENUM_CREATORROLE_AUTHOR???           
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1External Organizations, ou_persistent22              
2Potsdam Institute for Climate Impact Research, ou_persistent13              

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 ???ViewItemFull_lblAbstract???: The Helmholtz decomposition splits a sufficiently smooth vector field into a gradient field and a divergence-free rotation field. Existing decomposition methods impose constraints on the behavior of vector fields at infinity and require solving convolution integrals over the entire coordinate space. To allow a Helmholtz decomposition in, we replace the vector potential in R³ by the rotation potential, an n-dimensional, antisymmetric matrix-valued map describing n(n-1)/2 rotations within the coordinate planes. We provide three methods to derive the Helmholtz decomposition: (1) a numerical method for fields decaying at infinity by using an n-dimensional convolution integral, (2) closed-form solutions using line-integrals for several unboundedly growing fields including periodic and exponential functions, multivariate polynomials and their linear combinations, (3) an existence proof for all analytic vector fields. Examples include the Lorenz and Rössler attractor and the competitive Lotka–Volterra equations with n species.

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???ViewItemFull_lblLanguages???: eng - English
 ???ViewItemFull_lblDates???: 2022-03-152023-02-232023-09-15
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 ???ViewItemFull_lblIdentifiers???: ???ENUM_IDENTIFIERTYPE_DOI???: 10.1016/j.jmaa.2023.127138
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???ViewItemFull_lblSourceTitle???: Journal of Mathematical Analysis and Applications
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???ViewItemFull_lblPages???: ???lbl_noEntry??? ???ViewItemFull_lblSourceVolumeIssue???: 525 (2) ???ViewItemFull_lblSourceSequenceNo???: 127138 ???ViewItemFull_lblSourceStartEndPage???: ???lbl_noEntry??? ???ViewItemFull_lblSourceIdentifier???: ???ENUM_IDENTIFIERTYPE_CONE???: https://publications.pik-potsdam.de/cone/journals/resource/1096-0813
???ENUM_IDENTIFIERTYPE_PUBLISHER???: Elsevier