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arxiv: 2606.07501 · v1 · pith:L7IRLSRNnew · submitted 2026-06-05 · 🧮 math.AP

On the non-uniqueness of solutions of the axi-symmetric swirl-free Navier-Stokes equations, I

Alexandru D. Ionescu , Hao Jia , Stan Palasek This is my paper

Pith reviewed 2026-06-27 21:18 UTC · model grok-4.3

classification 🧮 math.AP
keywords Navier-Stokes equationsself-similar solutionsaxial symmetryswirl-freenon-uniquenessunstable modesnumerical construction
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The pith

New unstable self-similar solutions to the 3D Navier-Stokes equations exist in the axially symmetric swirl-free class.

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

This paper constructs numerically a new class of self-similar solutions to the incompressible Navier-Stokes equations in three dimensions. The solutions are axially symmetric, swirl-free, and homogeneous of degree minus one at infinity, with global pointwise approximation errors of order 10 to the minus 10. Linearization around the constructed profiles reveals unstable modes. A sympathetic reader cares because earlier solutions of this type have been used to establish non-uniqueness results, and the new examples sit inside the restricted axially symmetric swirl-free function space.

Core claim

In this paper we construct numerically a new class of unstable self-similar solutions of the incompressible Navier-Stokes equations in R^3. Our solutions are axially symmetric and homogeneous of degree -1 at infinity, and are unstable in the sense that the linearization around these solutions contains unstable modes. The main novelty is that we discover the existence of such solutions in the space of axially symmetric swirl-free vector fields. These approximate solutions are defined on all of R^3 and achieve global pointwise residuals of order 10^{-10}.

What carries the argument

Numerically constructed approximate solutions in the axially symmetric swirl-free class together with the spectrum of the linearized operator around them.

If this is right

  • The new solutions supply candidates for proving non-uniqueness of Navier-Stokes solutions inside the axially symmetric swirl-free class.
  • The presence of unstable modes indicates that these profiles can serve as building blocks for non-uniqueness constructions analogous to those based on earlier numerically discovered solutions.
  • The construction demonstrates that such solutions can be found without swirl, narrowing the function space in which non-uniqueness may hold.

Where Pith is reading between the lines

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

  • Confirmation that the profiles are exact would allow a direct route to non-uniqueness statements that avoid additional symmetry assumptions.
  • The same numerical approach could be tested on solutions with different homogeneity degrees or in spaces that include swirl to map the boundary between uniqueness and non-uniqueness.
  • If the unstable modes survive rigorous analysis, they may connect to questions about possible singularity formation in the swirl-free setting.

Load-bearing premise

The functions computed with pointwise residuals of order 10^{-10} are close enough to exact solutions that the spectrum of the linearized operator around them correctly identifies genuine unstable modes of the nonlinear problem.

What would settle it

A higher-precision numerical refinement or rigorous existence proof that removes the unstable modes from the spectrum while keeping the residual below 10^{-10}.

Figures

Figures reproduced from arXiv: 2606.07501 by Alexandru D. Ionescu, Hao Jia, Stan Palasek.

Figure 1
Figure 1. Figure 1: Plots of the solutions (Ψ, Ω) and (Ψe, Ω) described in the Main Nu- e merical Result 1.3 restricted to domain (r, z) ∈ [0, 50]2 . Remark 1.4. Our solutions (Ψ, Ω), (Ψe, Ω) e are represented as B-splines of degree 10 with about 148,000 degrees of freedom for each variable. We use global L∞ norms evaluated on dense grids in (1.18) to validate these solutions (see also [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Plots of the physical velocity fields (Ur, Uz) and (Uer,Uez) correspond￾ing to the solutions (Ω, Ψ) and (Ωe, Ψ), restricted to domain ( e r, z) ∈ [0, 50]2 . This numerical result strongly indicates that the approximate solutions constructed above can be upgraded to exact solutions. Specifically, the main objective of our program is to prove rigorously that there exist exact solutions (Ψ, Ω) ∈ C 2 e,o(R 2 )… view at source ↗
Figure 3
Figure 3. Figure 3: The coordinate mapping (u, v) 7→ (r, z) for (A, p) = (50, 4). Left: a uniform grid in the computational domain (u, v) ∈ [0, 50]2 . Right: the image of the same grid lines in the physical domain (r, z), showing the concentration of resolution near the origin. In terms of the new variables, the equations (1.15) are equivalent to the system LuvΨ2 − Ω2 = 0, LuvΩ2 +  K(u)∂u + K(v)∂v + 2 Ω2 + 2 2K(u)K(v) [PIT… view at source ↗
Figure 4
Figure 4. Figure 4: Evolution of the nonlinear solution (σ/κ)(Ψ′ 2 , Ω ′ 2 ) on the computa￾tional domain (u, v) ∈ [0, 50]2 , plotted for three values σ = 50, 100, 150. a Newton-Krylov method (GMRES). The solver is preconditioned by a sparse direct solver (SuperLU) applied to the Jacobian matrix, which is computed at the beginning of the step and lagged for subsequent inner iterations. The continuation step size δσ adapts dyn… view at source ↗
Figure 5
Figure 5. Figure 5: The two highest eigenvalues detected by the solver during the σ￾continuation. Both curves represent real single eigenvalues. that one single real eigenvalue becomes positive (unstable) at σ ≈ 103, and grows steadily as σ increases towards 160. The eigenvalue solver detects one other real eigenvalue that increases towards 0, but never crosses the axis before σ reaches 160. See [PITH_FULL_IMAGE:figures/full… view at source ↗
Figure 6
Figure 6. Figure 6: Knot density (knots per unit of u or v) of the splines used in the refinement stage, showing an adaptive multi-zone grid of degree-10 B-splines, concentrating up to ∼ 50 knots per unit in the regions where the solution gradi￾ents are steepest, and tapering to ∼ 2 knots per unit in the far field. (SuperLU). The result is a high-precision solution of the system (2.3) with a residual norm below 3 × 10−11 at t… view at source ↗
Figure 7
Figure 7. Figure 7: Residual heatmaps of the solutions Ψ, Ω, Ψ, e Ω in the physical space e R 2 (compare with the main bounds (1.18) and (1.20)). Notice that the residuals of the stream functions Ψ, Ψ are more significant in the gluing region, while the e residuals of the vorticities Ω, Ω are largest near the point ( e r, z) = (26, 0). The residuals in the gluing region can be further reduced by improving numerically the ansa… view at source ↗
Figure 8
Figure 8. Figure 8: The leading eigenvalue during 10 different simulations, using several methods, computed in either the physical domain (PD), the computational do￾main (CD), or the rescaled physical domain (RPD). The plotted curve actually consists of 10 nearly indistinguishable overlapping curves. The maximum devia￾tion of the leading eigenvalue we have observed among all of these simulations is less than 5 × 10−3 , for al… view at source ↗
Figure 9
Figure 9. Figure 9: illustrates the convergence properties of both the nonlinear base solution (Ψ, Ω) and the principal unstable eigenmode (Ψe, Ω), as we track the maximum absolute residual and the e eigenvalue deviation |λ − λref| as a function of the total degrees of freedom (DOF), using B￾splines of degrees 6, 8, and 10. The results exhibit clear spectral-like convergence. For instance, transitioning from degree-6 to degre… view at source ↗
read the original abstract

In this paper we construct numerically a new class of unstable self-similar solutions of the incompressible Navier-Stokes equations in $\mathbb{R}^3$. Our solutions are axially symmetric and homogeneous of degree $-1$ at $\infty$, and are unstable in the sense that the linearization around these solutions contains unstable modes. Solutions of this type have been discovered numerically by Guillod and \v{S}ver\'ak and Hou, Wang, and Yang, and have applications to proving non-uniqueness results. The main novelty in this paper is that we discover the existence of such solutions in the space of axially symmetric swirl-free (ASSF) vector fields. These approximate solutions are defined on all of $\mathbb R^3$ and achieve global pointwise residuals of order $10^{-10}$. We discuss the numerical construction of these solutions in detail, as well as their relevance to the problem of non-uniqueness of solutions of the incompressible Navier-Stokes equations in 3D, in the space of ASSF solutions.

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 / 2 minor

Summary. The paper claims to numerically construct a new class of axially symmetric swirl-free (ASSF) self-similar solutions to the 3D incompressible Navier-Stokes equations that are homogeneous of degree -1 at infinity. These approximate solutions are defined on all of R^3, achieve global pointwise residuals of order 10^{-10}, and the linearization around them is shown to contain unstable modes; the work discusses the numerical construction in detail and its relevance to non-uniqueness results in the ASSF class, extending prior numerical examples by Guillod-Šverák and Hou-Wang-Yang.

Significance. If the approximations are close enough to exact solutions that the reported unstable modes persist for the true linearization, the result would supply the first explicit numerical examples of unstable ASSF self-similar profiles, furnishing concrete candidates that could be used in future analytic arguments for non-uniqueness of weak solutions to the axi-symmetric swirl-free Navier-Stokes system.

major comments (1)
  1. [Abstract and numerical-construction section] Abstract and numerical-construction section: the central claim that the linearization around the computed profiles contains unstable modes rests on the assumption that pointwise residuals of order 10^{-10} produce only negligible perturbations to the spectrum. No a-posteriori eigenvalue error bound, resolvent estimate, or mesh-refinement/continuation argument is supplied to control the distance between the spectrum of the discrete linearized operator and that of the exact (unknown) solution; because eigenvalues near the imaginary axis can shift by O(1) under small but structured perturbations when the resolvent is ill-conditioned, this gap directly affects the reliability of the instability conclusion.
minor comments (2)
  1. [Linearization discussion] The manuscript should clarify the precise function space in which the linearization is considered (e.g., weighted Sobolev spaces adapted to the homogeneity -1 decay) and state the precise definition of “unstable mode” (growth rate, multiplicity, etc.).
  2. [Numerical results] Figure captions and tables reporting residuals should include the precise discretization parameters (mesh size, truncation radius, solver tolerance) so that the 10^{-10} figure can be reproduced.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful and constructive report. The single major comment is addressed point-by-point below.

read point-by-point responses

  1. Referee: [Abstract and numerical-construction section] Abstract and numerical-construction section: the central claim that the linearization around the computed profiles contains unstable modes rests on the assumption that pointwise residuals of order 10^{-10} produce only negligible perturbations to the spectrum. No a-posteriori eigenvalue error bound, resolvent estimate, or mesh-refinement/continuation argument is supplied to control the distance between the spectrum of the discrete linearized operator and that of the exact (unknown) solution; because eigenvalues near the imaginary axis can shift by O(1) under small but structured perturbations when the resolvent is ill-conditioned, this gap directly affects the reliability of the instability conclusion.

    Authors: The referee is correct that the manuscript supplies no explicit a-posteriori eigenvalue error bound, resolvent estimate, or quantitative mesh-refinement/continuation argument controlling the distance between the discrete and exact spectra. While the reported pointwise residuals of order 10^{-10} and our internal mesh-refinement checks indicate robustness of the observed unstable eigenvalues, these checks are not presented in a form that rigorously bounds possible O(1) shifts. We will therefore revise the numerical-construction section to include a perturbation analysis that uses the residual size together with a numerical estimate of the resolvent norm on a contour separating the unstable eigenvalues from the rest of the spectrum. revision: yes

Circularity Check

0 steps flagged

No circularity: direct numerical construction of approximate solutions

full rationale

The paper reports a numerical search for approximate self-similar ASSF solutions of the Navier-Stokes equations, achieving pointwise residuals of order 10^{-10} on all of R^3, followed by linearization to detect unstable modes. No derivation chain, equation, or claim reduces a result to a fitted parameter defined from the same data, a self-citation load-bearing premise, or any of the enumerated circular patterns. The central output is the computed profiles themselves; the instability statement is an analysis performed on those profiles rather than a prediction forced by construction. This is a standard numerical existence result with no self-referential reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The paper is a numerical construction report; the abstract introduces no new mathematical axioms, free parameters fitted to data, or postulated entities beyond the standard incompressible Navier-Stokes framework.

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Reference graph

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