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Engheta et al. found such an effect in free space in 1980, published as

On the Near-Zone Inverse Doppler Effect August 1980 IEEE Transactions on Antennas and Propagation 28(4)

Yehuda Ben-Shimol and Dan Censor found none in a two dimensional case in 1998, published as

Contribution to the problem of near-zone inverse Doppler effect, Radio Science, Volume 33, Number 3, pages 463-474, May-June 1998

The near zone by itself has no Doppler shift since it is non retarded, which seems similar to Ben-Shimol—Censor.

If the combined fields of near fields plus far fields is seen as a whole then, when changing distance to a source, an inverse Doppler effect can be observed since the near fields starts to dominate in strength over the far fields close to the source and thus opposing the traditional Doppler effect of the far fields. This view is similar to Engheta et al. No net inverse Doppler effect can exist though. It might better be described as a convective effect maybe.

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    $\begingroup$ "The question is of critical importance for the vacuum field energy" care to elaborate a bit more on that line of reasoning? $\endgroup$
    – lurscher
    Commented Sep 20, 2019 at 23:24
  • $\begingroup$ I now read that the near fields are retarded as well. What is the case? $\endgroup$ Commented Sep 2, 2021 at 8:01

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Although I'm far from this field of research for more than 20 years now, I have a small comment.

May I suggest that you refer to the full paper supporting part of Engheta et al. findings for inverse Doppler effect for the Electrical field and none for the Magnetic field. In addition, Engheta et al. discussed the case of a 3D oscillating dipole while in [1] we were discussing a 2D dipole.

best

yehuda

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    $\begingroup$ I updated your reference now. $\endgroup$ Commented Oct 10, 2019 at 22:14
  • $\begingroup$ Please update your answer as a response to my changes. $\endgroup$ Commented Nov 6, 2019 at 19:00

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