"Has anyone checked the work?"

tl;dr: no, probably not.

(I am aware of informal comments which have raised questions about whether the disk is a singularity, which would destroy the possibility of solving an initial value problem, unlike already-in-use approaches. Additionally, the rotating disk developed in the paper is clearly not present in non-axisymmetric elliptical galaxies dominated by radial motion, and so cannot replace dark matter in them; the paper only deals with disk-like approximations of spiral and axially-rotating spheroidal galaxies. There are plenty of galaxies where there's no common rotational axis, but there's still a rotational curve problem for stars and hydrogen gas clouds moving inwards vs outwards. Lastly, the paper only claims to be a good approximation in the limit of weak gravitational fields, so very dense galaxy clusters (which will include the future collision of the Andromeda galaxy with our own) are not covered by the work in this paper: it makes no claim to be able to predict the outcome of that collision, which breaks the Vlasov condition. When you collide the dust and gas in these galaxies, or galaxies like them in our sky, you will see lots and lots of X-Rays and the like, while some fraction of actually collisionless matter would deform the galaxy and red/blueshift its component spectra.)

This is the work in question, fed into Google Scholar (sauce for the goose as for the gander: Springer adds Google Scholar links to each of the author's references) using the first URL you supply:

https://scholar.google.com/scholar?hl=en&as_sdt=0%2C5&q=http...

We then hit "Cited by 3"

https://scholar.google.com/scholar?cites=8505897096354068536...

Not much there, and two are author self-cites, with no citations on the newer papers, and no collaborators on these papers.

Glancing briefly through the two newer ones, I got distracted by one inconsistency which strikes me as glaring because the "novel form" (author's words) the author builds strongly hangs off it: I found "pseudo tensor", "pseudotensor" and "pseudo-tensor" at the very least, and promptly gave up reading more deeply. Choose just one, please, or don't try to use them at all [1].

Aside: EPJ+ charges authors a USD 3280 fee to publish each article. It's also not a journal working cosmologists or extragalactic astrophysicsts would follow closely.

Fortunately, readers without institutional access can find (again, via Google Scholar, at the link above) essentially the same material on researchgate (which says nothing either way about quality) so one can glance without handing Springer $30+ for the two newer articles. Even more fortunately, the link you supplied is open-access, and can be read there (the PDF is nicely formatted) without paying a fee.

Finally on this point, we can look the author up and see numerous papers with collaborators in (mainly terrestrial applications of) plasma physics, but the only papers on astrophysics (in the broadest sense) are those three most recent ones.

https://scholar.google.com/scholar?hl=en&as_sdt=0%2C5&q=%22G...

My comments above are nothing at all like "checking the work", but rather an excuse for why I personally would be in no rush to do so.

It was however momentarily amusing that a partial theory designed to eliminate a large amount of cold collisionless non-radiating matter in a galaxy (the dark matter component) models the entire galaxy, stars and all as a cold collisionless non-radiating dust (whose individual "motes" generate the Vlasov fields which govern the dust's motion), and that this idea has excited some dark matter self-styled sceptics.

Lastly, the idea is not stupid. If we try to represent gravitational interactions as particle exchanges, there will be charges and potentials, and that this can strongly resemble electromagnetism has been intriguing physicists for many decades. The author is exploring a flavour of which, where the sources and potentials are similar to what is familiar to a plasma physicist. However, the writeups of this idea, found at the links above, are simply not general enough for our diverse zoo of real galaxies. It is also not general in scale: it's not useful at solar system scales (we are probably eventually interetsed in the behaviour of dark matter in things like the Hulse-Taylor system, or black hole binaries, etc.), at galaxy cluster scales, or for understanding the small fluctuations in the cosmic microwave background and their apparent connection to large scale structure.

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[1] Tensors (not pseudo-tensors) are enormously useful in General Relativity, because a tensor solution solved in one set of coordinates is solved in all systems of coordinates (including no coordinates at all). Matter should be specified as tensors. This has been done (e.g. the (Faraday) https://en.wikipedia.org/wiki/Electromagnetic_tensor). This lets us arrive at an understanding that works for you standing on your part of Earth using local notions of up, down, left, right, forward backward; and me standing somewhere else on Earth using my local notions of the same directions. It also works for people in the ISS who have to pick a conventional up/down, and whose local clocks tick fast (from our standing-on-Earth's-surface perspective). It also works for coordinates covering the solar system, the Milky way, or the entire cosmos. But not all physical systems have fully developed tensor theories. For them we may choose to introduce a pseudo-tensor, which are valid only for certain coordinate systems. A solution in one of those will work for some other systems of coordinates but not all, and if one is not careful in choosing coordinates, one can get things spectacularly wrong, like wildly wrong recoveries of the components of the energy-momentum 4-vector ("In some coordinate systems you think there is energy present when there is no energy in other coordinate systems" is a common symptom, and in this context means "you think you do not need a generator of curvature beyond the dust encoded in the pseudo-tensor"). If one is inconsistent about the spelling of pseudo-tensor, I think that's a bad sign about the extreme care in keeping different coordinate systems mutually and fully consistent when using them.

One runs into pseudo-tensor inconsistencies fairly often in cosmological contexts, so much so that it was dealt with in a USENET sci.physics FAQ entry: https://math.ucr.edu/home//baez/physics/Relativity/GR/energy...