The central result is global: within the paper’s specified class, a sufficiently small perturbation of an expanding affine solution remains small for all time and admits a unique global strong solution. Here, “strong” means a solution with the regularity required for the equations and the proof’s estimates. The conclusion is about controlled initial data, not arbitrary disturbances.
The paper also asks what happens when viscosity is taken to zero. It states that, on any fixed finite time interval, the viscous perturbation converges in the theorem’s stated function space to the corresponding perturbation for the compressible Euler equations. This is the inviscid limit: the viscous model approaches the version without viscosity.
Turning a moving boundary into a fixed problem
The equations involve a free boundary, so the region occupied by the flow changes with time. The authors handle that geometry by rewriting the free-boundary equations in Lagrangian coordinates—variables that follow the motion—and placing the perturbation problem on a fixed domain. That change of description gives the proof a fixed setting in which to build its energy estimates.
The global construction is carried out in the special degenerate-viscosity case δ=γ, where the viscosity law is tied to the exponent used by the equations, and around the affine background. That equality is one of the result’s main boundaries: the supplied theorem does not establish the same construction for other viscosity exponents.
What the assumptions buy
The theorem’s assumptions are explicit: γ must be greater than 1; a specified combination of the two viscosity coefficients must lie in a bounded range; the affine background must satisfy the paper’s stated compatibility condition; and the initial perturbation variables η0 and η1 must have the required even extensions. The initial energy must also be small, written E_in≤ε̄.
The payoff is uniform control. In the displayed estimates, the generic constant C is independent of the viscosity coefficients, the time T, and the weighting parameters r1 and r2; a separate constant C_r1 may depend on r1. This uniformity underpins the zero-viscosity argument.
The stability statement is Lyapunov stability, a norm-based mathematical statement that an initially small perturbation remains small for all time. It describes the behavior of η within the model and its prescribed function spaces; it is not a general claim about every compressible flow.
The limit as viscosity fades
For the inviscid limit, the proof combines viscosity-uniform energy estimates with a standard compactness method. In practical terms, it keeps the relevant bounds from depending on the shrinking viscosity coefficients and uses them to identify the Euler limit. The theorem is stated for any T>0, but that means each fixed finite interval, not uniform convergence over all time.
The paper gives an explicit rate as well as convergence. Its difference-energy estimate is proportional to C_T times the square of the viscosity combination 2 mu-bar + lambda-bar, multiplied by the initial energy E_in. The bound therefore has quadratic scaling in that combination, while its constants and assumptions remain theorem-dependent.
A theorem with tight boundaries
The result has a tight scope. It is built for a spherically symmetric free-boundary problem and a prescribed class of perturbations; it does not establish global solutions for general nonspherical geometries or arbitrary initial data. The proof also centers on δ=γ and small initial energy, leaving other viscosity exponents and large perturbations outside the stated result.
That leaves several mathematical questions open: whether the construction can be extended beyond spherical symmetry, whether it survives when δ≠γ or the initial perturbation is larger, and whether the inviscid convergence can be made uniform on unbounded time intervals. None of those extensions is established in the supplied theorem.
The supplied document is an arXiv version 1 preprint in math.AP dated 26 August 2026. Its acknowledgements report Hong Kong PhD Fellowship Scheme support for the first author, General Research Fund support from Hong Kong’s RGC for the second author, and National Natural Science Foundation of China support for the third.
Paper data and sources
Original title: Expanding solutions to the compressible Navier-Stokes equations with degenerate viscosities in spherical symmetry: global existence and inviscid limit
Authors: Shuying Hu, Zhouping Xin, Yuan Yuan
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text