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arxiv: 2606.02517 · v1 · pith:NC3LIMULnew · submitted 2026-06-01 · 🧮 math.AP

Weak-strong uniqueness and low Mach number limit for a viscous compressible fluid around a rotating body

Thomas Eiter , \v{S}\'arka Ne\v{c}asov\'a , Florian Oschmann This is my paper

Pith reviewed 2026-06-28 13:26 UTC · model grok-4.3

classification 🧮 math.AP
keywords weak-strong uniquenesslow Mach number limitcompressible Navier-Stokesrotating bodyexterior domainrelative energy inequalityincompressible limit
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The pith

A relative energy inequality for weak solutions establishes weak-strong uniqueness and the low Mach number limit to incompressible rotating flow.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The authors derive a relative energy inequality for weak solutions of the isothermal compressible Navier-Stokes system in an exterior domain around a rotating body. This inequality is used to prove a weak-strong uniqueness principle, ensuring that weak solutions coincide with strong ones when the latter exist. It also shows that the compressible solutions converge to solutions of the incompressible Navier-Stokes equations for rotating flow as the Mach number tends to zero. These results apply to time-independent rigid body motion with decay conditions at infinity. Readers interested in fluid dynamics would care because the results justify using simpler incompressible models for low-speed compressible flows around moving bodies.

Core claim

The central claim on the paper's own terms is that a relative energy inequality holds for weak solutions to the compressible fluid equations around a rotating body, from which both a weak-strong uniqueness result and the low Mach number limit to the incompressible rotating flow follow.

What carries the argument

The relative energy inequality for weak solutions in the exterior domain, which serves as the basis for both uniqueness and the limit passage.

If this is right

  • If a strong solution exists, it is the unique weak solution.
  • The compressible system is well approximated by the incompressible rotating Navier-Stokes equations at low Mach numbers.
  • The results hold for time-independent rigid motions of the body with appropriate decay at spatial infinity.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The derivation technique might extend to time-dependent rotations if similar energy estimates can be obtained.
  • These results could inform the design of numerical schemes that transition between compressible and incompressible regimes.
  • Connections to other exterior domain problems in fluid mechanics may be explored using similar relative energy methods.

Load-bearing premise

The relative energy inequality can be derived for weak solutions of the compressible system in the exterior domain with the prescribed rigid motion of the body.

What would settle it

Observing a weak solution that differs from a known strong solution or does not approach the incompressible limit as the Mach number decreases would falsify the uniqueness and convergence claims.

read the original abstract

We study the flow of an isothermal compressible Newtonian fluid around a body that performs a (time-independent) rigid motion. We derive a weak-strong uniqueness principle, and show that in the low Mach number limit, the governing equation is well approximated by the Navier-Stokes equations for incompressible rotating flow. Both results are based on the derivation of a relative energy inequality for weak solutions to this exterior-domain problem.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The paper studies isothermal compressible Newtonian fluid flow around a body in time-independent rigid motion. It claims to derive a relative energy inequality for weak solutions in the exterior domain, from which a weak-strong uniqueness principle follows and the low Mach number limit is shown to approximate the incompressible rotating Navier-Stokes equations.

Significance. If the relative energy inequality is rigorously derived and holds under the stated conditions (exterior domain, rigid motion, decay at infinity), the results would extend relative-energy techniques to a technically challenging setting and provide useful uniqueness and approximation statements for compressible exterior flows with rotation.

major comments (1)
  1. [Abstract / relative energy inequality derivation] The abstract states that both the weak-strong uniqueness and the low Mach limit rest on the derivation of a relative energy inequality for weak solutions. The handling of the time-independent rigid body motion, the exterior-domain terms, and the decay conditions at spatial infinity in this derivation cannot be verified from the provided text; this is load-bearing for the central claims.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful review and the constructive comment. We address the major point below and are prepared to revise the manuscript accordingly to improve clarity.

read point-by-point responses

  1. Referee: [Abstract / relative energy inequality derivation] The abstract states that both the weak-strong uniqueness and the low Mach limit rest on the derivation of a relative energy inequality for weak solutions. The handling of the time-independent rigid body motion, the exterior-domain terms, and the decay conditions at spatial infinity in this derivation cannot be verified from the provided text; this is load-bearing for the central claims.

    Authors: The relative energy inequality is derived in Section 3. The time-independent rigid motion is incorporated via a Galilean-type change to the body-fixed frame, which produces additional linear and quadratic terms in the momentum equation; these are absorbed into the relative energy by adding suitable correctors that exploit the time-independence. The exterior-domain setting is treated by working in weighted Sobolev spaces (with weights ensuring integrability at infinity) and by localizing the estimates away from the body. Decay at spatial infinity is used to justify that all surface integrals over large spheres vanish in the limit, via the assumed integrability of the weak solutions and the structure of the relative energy. We acknowledge that the presentation of these steps may not be sufficiently explicit for verification and will revise the manuscript to expand the derivation, add intermediate estimates, and include explicit references to the handling of each term. revision: yes

Circularity Check

0 steps flagged

No significant circularity; relative energy inequality derived independently

full rationale

The paper's central results (weak-strong uniqueness and low Mach limit) rest on deriving a relative energy inequality for weak solutions of the compressible system in an exterior domain with rigid body motion and decay at infinity. This derivation is presented as a new technical step without reduction to fitted inputs, self-definitional loops, or load-bearing self-citations that force the outcome by construction. The abstract and reader's assessment confirm the inequality is obtained from the governing equations and assumptions, making the subsequent uniqueness and limit results logically downstream rather than tautological. No quoted steps exhibit the enumerated circular patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard domain assumptions in compressible fluid PDE theory and the validity of the relative energy approach in the rotating exterior setting. No free parameters or invented entities are indicated in the abstract.

axioms (2)
  • domain assumption Existence of weak solutions to the isothermal compressible NS system in the exterior domain with rigid rotation
    The uniqueness result applies to weak solutions, so their existence is presupposed.
  • standard math The relative energy functional satisfies standard inequalities under the given boundary and far-field conditions
    The paper derives the inequality from this background property.

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Reference graph

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