A rotating ball of gas held together by its own attraction sounds like it should have one fate. Either the spin flings it apart or gravity wins. Samuel R. Charles, in a 53-page preprint posted to arXiv on 30 July 2026, shows that the honest answer is neither: for a family of idealized stars governed by the compressible Euler-Riesz equations, rotation can steady the star or wreck it, and which one you get depends on a single mathematical dial describing how sharply the attractive force blows up when two bits of matter get close.
The equations in question describe a compressible fluid whose parts pull on each other through what mathematicians call a Riesz interaction: a force law that falls off with distance according to a tunable power. Newtonian gravity is one member of this family. Others show up in models of plasmas and in mathematical biology, where cells or organisms drift toward each other under a similar attraction. That shared structure is why the Euler-Riesz system gets studied as a single object rather than as three separate physics problems.
Charles calls the objects at the center of the paper rotating Riesz stars: steady states of the attractive Euler-Riesz equations that spin, rather than sitting still. The non-rotating versions have been studied for some time. Adding rotation changes both what exists and what survives a nudge.
Two regimes, two fates
The paper splits along a line that specialists call mass criticality. In the mass-subcritical regime, roughly the setting where the attraction is gentle enough that a configuration of fixed total mass can settle into a preferred shape, Charles proves two things. Rotating Riesz stars exist, provided the star's angular momentum profile (how its spin is distributed from the axis outward) satisfies a subhomogeneity condition, meaning it grows slowly enough as you move out. And those stars are nonlinearly stable. Nonlinear stability is a strong statement: perturb the star by a small but genuinely finite amount, not just an infinitesimal one, and it stays close to where it started rather than drifting off.
The proof runs through a technique called concentration compactness, which asks whether a sequence of configurations that get closer and closer to the best possible energy actually converges to something, or instead leaks away. Charles had to adapt the standard argument to the axisymmetric setting, and rotation introduced a failure mode with no counterpart in the non-rotating problem. The near-optimal configurations can stay perfectly well concentrated while migrating outward along rings whose radii run off to infinity. Nothing spreads out and nothing vanishes; the mass just travels. Ruling that out is one of the paper's technical demands.
In the mass-supercritical regime, where the attraction is more singular, the picture inverts. Working with polytropic pressure laws, a standard idealization in which pressure is a fixed power of density, Charles proves existence again, this time under a superhomogeneity assumption on the angular momentum profile. That result extends what was known past the small angular velocity regime, so it covers stars spinning at rates earlier arguments could not reach. The existence proof leans on mass-preserving scalings, ways of stretching and squeezing a configuration while keeping its total mass fixed, and the author notes these are more delicate to handle once rotation is present.
Then comes the reversal. By examining how the free energy behaves along those same scalings, and specifically finding it concave, Charles establishes that the mass-supercritical rotating Riesz stars are unstable. They exist, but they will not stay.
Why it matters
The headline result is the contrast, not either half alone. Rotation is often treated in physical intuition as a stabilizing influence, something that props a star up against collapse. This work shows that within one clean mathematical family, the sign of that effect flips depending on how singular the interaction is. Stabilizing in one regime, destabilizing in the other, with no change to the equations other than the exponent in the force law.
It is worth being clear about what this is. These are theorems about a system of partial differential equations, not observations of a star, and the model is an idealization: no magnetic fields, no radiation, no nuclear burning, no relativity. The value lies in rigor. Numerical simulation can suggest that a configuration will hold together, but a proof of nonlinear stability says that no perturbation of the allowed size can break it, and that is a different order of knowledge.
There is also a practical reason mathematicians care about the Riesz family in the aggregate. Proving something for the whole tunable range at once, rather than for Newtonian gravity alone, means the plasma physicists and the biological modelers get the result too. And the paper is a preprint, posted without peer review yet, so the arguments across those 53 pages still await the scrutiny of other specialists.