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arxiv: 2605.04386 · v1 · submitted 2026-05-06 · 🧮 math.AP

Global spherically symmetric solutions to the multidimensional isentropic compressible Navier--Stokes--Korteweg system with large initial data

Zhengzheng Chen , Fanfan Jiang This is my paper

Pith reviewed 2026-05-08 17:55 UTC · model grok-4.3

classification 🧮 math.AP
keywords Navier-Stokes-Korteweg systemspherically symmetric solutionsglobal existencelarge initial dataexterior domainisentropic compressible flowcapillary effects
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The pith

Global existence and uniqueness of spherically symmetric strong solutions holds for the isentropic compressible Navier-Stokes-Korteweg system with large initial data under suitable restrictions on the parameters alpha, beta and gamma.

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

The paper establishes global existence and uniqueness of spherically symmetric strong solutions to an initial-boundary value problem for the multidimensional isentropic compressible Navier-Stokes-Korteweg system in an unbounded exterior domain, even when the initial data is large. The pressure follows a power law, while viscosity and capillarity coefficients are allowed to depend on density through power parameters alpha, beta and gamma. The result applies when those parameters satisfy suitable restrictions and is proved by combining a radically weighted energy method with the technique of Kanel. A reader would care because the result removes smallness assumptions on the data and handles capillary forces in a setting that models real fluids with surface tension effects.

Core claim

Under suitable restrictions on the parameters alpha, beta and gamma, the global existence and uniqueness of spherically symmetric strong solutions to the initial-boundary value problem of the multidimensional isentropic compressible Navier-Stokes-Korteweg system is established in an unbounded exterior domain for large initial data.

What carries the argument

The radically weighted energy method combined with the technique developed by Y. Kanel.

If this is right

  • Strong solutions remain globally regular without developing singularities in finite time.
  • The result covers both constant-viscosity cases and density-dependent viscosity cases with matching power laws.
  • Uniqueness holds among all spherically symmetric strong solutions.
  • The approach extends classical large-data techniques to systems that include capillary forces.

Where Pith is reading between the lines

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

  • Spherical symmetry plus radial weighting appears sufficient to control the multidimensional problem even without smallness on the data.
  • The same method may adapt to related systems such as non-isentropic or quantum Navier-Stokes models.
  • The parameter restrictions likely guarantee that the energy functional remains coercive and controls the density away from zero and infinity.

Load-bearing premise

The initial data must be spherically symmetric and have sufficient regularity in the unbounded exterior domain.

What would settle it

Construction of a spherically symmetric initial datum with large size that produces a strong solution blowing up in finite time when the parameters alpha, beta, gamma lie outside the allowed range.

read the original abstract

In this paper, we investigate the global existence of spherically symmetric strong solutions with large initial data to an initial-boundary value problem of the multidimensional isentropic compressible Navier-Stokes-Korteweg system in an unbounded exterior domain. We consider the case when the pressure $p(\rho)=\rho^\gamma$, the viscosity coefficients $\mu(\rho)$ and $ \lambda(\rho)$ satisfy either $\mu(\rho)=\tilde{\mu}, \lambda(\rho)=\tilde{\lambda}\rho^\alpha$ or $\mu(\rho)=\tilde{\mu}\rho^\alpha, \lambda(\rho)=\tilde{\lambda}\rho^\alpha$, and the capillarity coefficient $\kappa(\rho)=\tilde{\kappa}\rho^\beta$, where $\alpha,\beta,\gamma \in \mathbb{R}$ are parameters, and $\tilde{\mu},\tilde{\lambda},\tilde{\kappa}$ are given real constants. Under suitable restrictions on the parameters $\alpha,\beta$ and $\gamma$, we establish the global existence and uniqueness of spherically symmetric strong solutions. The proof relies on the radically weighted energy method combined with the technique developed by Y.~Kanel'\cite{28}.

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

0 major / 2 minor

Summary. The paper claims to establish the global existence and uniqueness of spherically symmetric strong solutions with large initial data for an initial-boundary value problem of the multidimensional isentropic compressible Navier-Stokes-Korteweg system in an unbounded exterior domain. The pressure is p(ρ)=ρ^γ; the viscosity coefficients satisfy either μ(ρ)=μ̃ (constant) and λ(ρ)=λ̃ ρ^α or both μ(ρ)=μ̃ ρ^α and λ(ρ)=λ̃ ρ^α; the capillarity coefficient is κ(ρ)=κ̃ ρ^β. Under suitable restrictions on the real parameters α, β, γ, the result follows from a radially weighted energy method combined with Kanel's technique.

Significance. If the a priori estimates close under the stated parameter restrictions, the result extends global strong-solution theory for the NSK system to large data in exterior domains while retaining spherical symmetry. The radially weighted energy functional is a natural device for controlling decay at infinity and the geometric (n-1)/r terms that arise after reduction to radial coordinates; pairing it with Kanel's 1D L^∞ technique for density is a standard and appropriate choice. The work therefore supplies a concrete, falsifiable existence theorem for a physically relevant regime (compressible flow with capillarity) without smallness assumptions on the data.

minor comments (2)
  1. The abstract states that the result holds 'under suitable restrictions on the parameters α, β and γ' but does not list the precise inequalities. Stating the admissible range explicitly (even if only by reference to the main theorem) would allow readers to assess the scope immediately.
  2. The notation for the constant coefficients (μ̃, λ̃, κ̃) is introduced in the abstract; a single sentence in the introduction confirming that these are fixed nonzero constants would remove any ambiguity about whether they may vanish or change sign.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive evaluation of our manuscript and the recommendation for minor revision. The assessment of the significance of extending global strong-solution theory for the NSK system to large data in exterior domains under spherical symmetry is appreciated. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper proves global existence and uniqueness of spherically symmetric strong solutions to the isentropic NSK system via reduction to radial symmetry, application of a radially weighted energy functional, and invocation of Kanel's external 1D technique for uniform density bounds. Parameter restrictions on α, β, γ close the a priori estimates without smallness assumptions on data. No step reduces by definition to the target result, no fitted quantities are relabeled as predictions, and the sole cited technique is from an independent source (Kanel' 1968) rather than a self-citation chain. The derivation is self-contained against external mathematical benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result rests on standard Sobolev-type embeddings, the applicability of Kanel's technique to this system, and the spherical symmetry reduction; no new entities are introduced.

axioms (2)
  • domain assumption The initial data is spherically symmetric and belongs to suitable Sobolev spaces allowing the weighted energy estimates to close.
    Invoked to reduce the multidimensional problem and control the radial weights.
  • domain assumption Kanel's technique extends directly to the NSK system with the given density-dependent coefficients.
    Cited without re-derivation in the abstract.

pith-pipeline@v0.9.0 · 5513 in / 1182 out tokens · 77899 ms · 2026-05-08T17:55:36.283338+00:00 · methodology

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