Here is a 3-paragraph summary of the blog post for mathopen.com:
Maxwell's equations, which govern electromagnetism, follow the rules of special relativity and remain unchanged under Lorentz transformations. Fluid dynamics, on the other hand, plays by the older rules of Newtonian mechanics and is invariant under Galilean transformations instead. When you try to combine these two frameworks to study electrically conductive fluids like plasma, the mathematical incompatibility creates a serious problem.
The key question the post explores is whether there is a way to study magnetohydrodynamics, the science of electrically conductive fluids, without dragging the full machinery of special relativity into the picture. The answer lies in finding a Galilean limit of Maxwell's equations, essentially a version of electromagnetism that has been adapted to play nicely with the slower, non-relativistic world of classical fluid flow.
It turns out there are actually two distinct Galilean limits of electromagnetism, not just one. These two limits correspond to different physical regimes and offer a cleaner, more tractable mathematical foundation for studying phenomena like plasma behavior. The post digs into the structure of these limits and what they reveal about the surprising richness hiding inside classical electromagnetism.