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arxiv: 2604.23915 · v2 · submitted 2026-04-26 · ⚛️ physics.flu-dyn · math.DS· math.OC

Improved global stability bounds for two-dimensional plane Poiseuille flow

Vicente Iligaray , Danilo Aballay , Federico Fuentes This is my paper

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

classification ⚛️ physics.flu-dyn math.DSmath.OC
keywords plane Poiseuille flowglobal stabilityLyapunov functionalssemidefinite programmingnonlinear stabilityenergy stabilityReynolds number bounds
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The pith

Quartic Lyapunov functionals certify global nonlinear stability of plane Poiseuille flow up to Reynolds number 106.8 at critical lengths.

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

The paper seeks to raise the proven threshold for nonlinear global stability of two-dimensional pressure-driven channel flow by building special quartic functionals that decrease along all trajectories. It does so by splitting velocity perturbations into a small number of energy eigenmodes plus a controlled tail, then converting the requirement that the functional decreases into a semidefinite program whose feasibility can be checked by computer. At the streamwise length where the classical energy method of Orr gives the lowest limit of roughly 87.59, the new functionals certify stability all the way to 106.8. This narrows the region where the laminar profile could still be destabilized by finite-amplitude disturbances and demonstrates that modest computational effort can improve century-old analytical bounds.

Core claim

The authors show that quartic Lyapunov functionals constructed from finite sets of energy eigenmodes, augmented with explicit bounds on the remaining infinite tail, satisfy the conditions for global stability when their associated semidefinite programs are feasible; this yields a certified global stability limit of approximately 106.8 at the critical energy-stable streamwise length where Re_E is approximately 87.59, constituting a 22 percent improvement over the 1907 energy bound.

What carries the argument

A quartic Lyapunov functional built from a finite 'mode set' of energy eigenmodes of the linearized operator together with explicit tail bounds, whose time derivative is rendered negative by semidefinite programming feasibility.

If this is right

  • The flow is globally stable to arbitrary perturbations up to Re ≈ 106.8 at the critical length, improving 22% on Re_E ≈ 87.59.
  • Mode sets as small as five modes are sufficient to capture the necessary nonlinear features for improved certificates.
  • The SDP-based certification method applies uniformly over the streamwise lengths examined and consistently exceeds the energy bound.
  • Explicit tail bounds allow the reduction of the infinite-dimensional stability problem to a finite semidefinite program.

Where Pith is reading between the lines

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

  • The technique could be applied to other parallel shear flows to obtain tighter nonlinear stability bounds without solving the full nonlinear problem.
  • Success with small mode sets indicates that the decay mechanisms after initial growth can be captured by low-dimensional projections.
  • Further refinement of mode selection or tail estimates might push the certified limit even closer to observed transition values.

Load-bearing premise

The finite mode sets chosen, together with the explicit bounds on the infinite tail, capture all essential nonlinear interactions so that feasibility of the semidefinite program implies the Lyapunov functional decreases for every perturbation.

What would settle it

A numerical simulation of the full Navier-Stokes equations at Re = 106.8 and the critical streamwise length, initialized with a perturbation whose energy grows to a finite value without decaying, would falsify the global stability certificate.

Figures

Figures reproduced from arXiv: 2604.23915 by Danilo Aballay, Federico Fuentes, Vicente Iligaray.

Figure 1
Figure 1. Figure 1: The left panel shows the energy eigenvalue branches of 2D Poiseuille flow at 𝑅𝑒 = 100 as a function of the streamwise wavenumber 𝛼, properly labelled by the parity of the corresponding streamfunction. Eigenvalues consistent with 𝐿 = 3 are labelled as (𝑛, 𝑘), which, when 𝑛 ≠ 0, correspond to the 𝑘-th largest eigenvalues at the fixed wavenumber 𝛼𝑛 = 2𝜋 𝐿 𝑛, and when 𝑛 = 0 correspond to 𝜆 = − 1 4𝑅𝑒 (𝑘 + 1) 2𝜋… view at source ↗
Figure 2
Figure 2. Figure 2: Global stability curves resulting from using the SOS-Lyapunov framework with the mode sets 𝒰𝑚 for 𝑚 = 5, 6, 7, 9, 11, 13 (see view at source ↗
Figure 3
Figure 3. Figure 3: The left panel shows the nearly indistinguishable feasibility thresholds using the mode set 𝒰9 (see view at source ↗
read the original abstract

This work provides new lower bounds on the global (nonlinear) stability limit of pressure-driven two-dimensional plane Poiseuille flow, improving on the energy stability limit, $Re_E$, originally computed by Orr in 1907. Using a computer we carefully construct quartic Lyapunov functionals of the velocity perturbations about the laminar profile, which certify the nonlinear stability of the flow to arbitrary perturbations. The formulation combines a decomposition of the velocity into finitely many energy eigenmodes, referred to as a 'mode set', and an infinite-dimensional 'tail', together with explicit bounds that recast the Lyapunov inequality conditions as semidefinite programs, whose feasibility is tested. Over the streamwise lengths considered, the certified stability limit exceeds the classical energy bound. In particular, at the critical energy-stable streamwise length, where $Re_E\approx 87.59$, the flow is found to be globally stable up to $Re \approx 106.8$ (representing a $22\%$ improvement). Various modestly-sized mode sets, capable of capturing sufficient features of the nonlinear dynamics of energy growth and subsequent decay, are proposed and found to be successful in producing improved bounds, with the simplest one involving only five modes.

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

2 major / 2 minor

Summary. The manuscript develops a computational framework for certifying improved lower bounds on the global nonlinear stability of two-dimensional plane Poiseuille flow. It constructs quartic Lyapunov functionals by decomposing perturbations into finite sets of energy eigenmodes plus an infinite tail, derives explicit bounds on the tail contributions, and recasts the Lyapunov decrease conditions as semidefinite programs (SDPs) whose feasibility certifies stability. At the critical streamwise length where the energy stability limit is Re_E ≈ 87.59, the method yields a certified global stability threshold of Re ≈ 106.8 (a 22% improvement), with successful results obtained from modestly sized mode sets including one with only five modes.

Significance. If the certificates hold, the work provides a concrete advance over the classical energy stability bound of Orr (1907) by delivering rigorous, computer-assisted lower bounds on the nonlinear stability threshold. The SDP reformulation of Lyapunov conditions is a standard rigorous technique that, when combined with explicit tail bounds, yields falsifiable stability certificates without fitting parameters to the target Reynolds number. The demonstration that small mode sets suffice is computationally attractive and may extend to other shear-flow problems.

major comments (2)
  1. [section deriving the tail bounds and SDP constraints] The central claim that the flow is globally stable up to Re ≈ 106.8 rests on the assertion that the chosen finite mode sets (including the five-mode set) plus the derived tail bounds together dominate all nonlinear interactions. The manuscript must explicitly verify that no quadratic or cubic term arising from tail self-interactions or cross-mode couplings is under-bounded; otherwise SDP feasibility certifies stability that may not exist in the infinite-dimensional system. This verification is load-bearing and should appear in the section deriving the tail bounds and the associated SDP constraints.
  2. [numerical implementation and verification subsection] Numerical details required for reproducibility and soundness are missing or insufficiently prominent: the SDP solver tolerances, the explicit values of the tail bounding constants, and the procedure used to confirm that the selected mode sets capture all essential nonlinear dynamics. Without these, it is impossible to assess whether the reported 22% improvement is robust or sensitive to truncation choices.
minor comments (2)
  1. The abstract states the improvement but does not mention the numerical tolerances or tail constants; adding one sentence on these would strengthen reader confidence without lengthening the abstract unduly.
  2. Ensure that every equation defining the Lyapunov functional, the tail estimates, and the SDP matrices is numbered and cross-referenced in the text.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful review and constructive feedback on our manuscript. We are pleased that the significance of the work is recognized. We respond to each major comment below, agreeing to enhance the manuscript with additional verifications and numerical details as suggested.

read point-by-point responses

  1. Referee: [section deriving the tail bounds and SDP constraints] The central claim that the flow is globally stable up to Re ≈ 106.8 rests on the assertion that the chosen finite mode sets (including the five-mode set) plus the derived tail bounds together dominate all nonlinear interactions. The manuscript must explicitly verify that no quadratic or cubic term arising from tail self-interactions or cross-mode couplings is under-bounded; otherwise SDP feasibility certifies stability that may not exist in the infinite-dimensional system. This verification is load-bearing and should appear in the section deriving the tail bounds and the associated SDP constraints.

    Authors: We thank the referee for highlighting this crucial aspect. The tail bounds are constructed precisely to provide rigorous upper bounds on all possible nonlinear terms involving the tail, including self-interactions and cross terms with the finite modes. This is achieved by using the orthogonality of the energy eigenmodes and applying appropriate inequalities to estimate the contributions from the infinite tail. However, to make this explicit as requested, we will add a dedicated paragraph or subsection in the revised manuscript that explicitly lists and bounds each type of quadratic and cubic term, demonstrating that none are under-estimated in the Lyapunov functional decrease condition. This will confirm the validity of the SDP certificates for the full system. revision: yes

  2. Referee: [numerical implementation and verification subsection] Numerical details required for reproducibility and soundness are missing or insufficiently prominent: the SDP solver tolerances, the explicit values of the tail bounding constants, and the procedure used to confirm that the selected mode sets capture all essential nonlinear dynamics. Without these, it is impossible to assess whether the reported 22% improvement is robust or sensitive to truncation choices.

    Authors: We agree that these numerical details are essential for assessing the robustness of the results. In the revised manuscript, we will expand the numerical implementation subsection to include: (i) the specific SDP solver employed and its tolerance settings, (ii) the computed values of the tail bounding constants for each mode set considered, and (iii) a description of the verification procedure, which involves checking that the finite mode sets include all modes with significant energy growth rates below the certified Re, supplemented by comparisons with direct numerical simulations of the linearized operator to ensure capture of essential dynamics. These additions will allow readers to reproduce and verify the sensitivity to truncation. revision: yes

Circularity Check

0 steps flagged

No circularity: SDP feasibility certificates are independent of target Re

full rationale

The paper derives improved global stability bounds by constructing quartic Lyapunov functionals over finite energy eigenmode sets plus explicit tail bounds, then recasting the resulting Lyapunov inequalities as SDPs whose feasibility is checked numerically. Feasibility at a given Re certifies nonlinear stability up to that Re without fitting parameters to the target stability limit itself; the certified Re (e.g., 106.8 at Re_E≈87.59) is simply the largest value for which the SDP remains feasible. No self-citation chain, self-definitional loop, or fitted-input-as-prediction appears in the derivation. The method is a computational certificate whose validity rests on the (explicitly stated) sufficiency of the chosen modes and bounds, not on circular reduction to the input data.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The claim rests on standard spectral decomposition of the linearized operator and validity of a priori bounds on the tail; no new physical entities are introduced.

free parameters (1)
  • Mode set selection and size
    Specific choices of which energy eigenmodes to retain in the finite set are made to achieve SDP feasibility and capture dynamics; these are not data-fitted constants but design choices affecting the resulting bound.
axioms (2)
  • standard math The linearized Poiseuille operator admits a complete set of energy eigenmodes allowing decomposition into finite set plus tail
    Invoked to justify splitting perturbations into mode set and infinite-dimensional tail for the Lyapunov construction.
  • domain assumption Explicit a priori bounds exist that control the tail contributions in the Lyapunov derivative
    Required to convert the infinite-dimensional Lyapunov inequality into a finite SDP.

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Reference graph

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