pith. sign in

arxiv: 2606.18232 · v1 · pith:KPBNLSS6new · submitted 2026-06-16 · ⚛️ physics.flu-dyn

Quartic Lyapunov functions for global fluid stability

David Darrow , Elizabeth Carlson , David Goluskin This is my paper

Pith reviewed 2026-06-26 22:22 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn
keywords Lyapunov functionsglobal stabilityshear flowsReynolds numberplane Couette flowplane Poiseuille flowfluid dynamicspolynomial optimization
0
0 comments X

The pith

A three-parameter family of quartic polynomials serves as Lyapunov functions to prove global stability of two-dimensional shear flows at Reynolds numbers beyond the reach of the energy method.

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

The paper establishes that a simple family of quartic polynomial Lyapunov functions can certify global stability for fluid flows where the classical energy method fails due to transient growth. By representing shear flows in complex variables, the authors exploit symmetries to shrink the search space and replace some numerical steps with analytical inequalities. This yields the simplest known non-quadratic Lyapunov functions for two-dimensional parallel shear flows. The functions are then used to verify global stability for plane Couette and plane Poiseuille flows at higher Reynolds numbers than the energy method allows. The work also shows how to prove stability over intervals of Reynolds numbers rather than at isolated values.

Core claim

We show how to exploit symmetries of shear flows via convenient complex variable representations, greatly reducing the problem size. We then refine key inequalities, replace several expensive computational steps with simpler analytical alternatives, and show how to prove global stability over a range of Reynolds numbers. Our analysis identifies the simplest class of non-quadratic Lyapunov functions for two-dimensional parallel shear flows: a three-parameter family of quartic polynomials. Using these Lyapunov functions, we verify global stability of 2-D plane Couette flow and plane Poiseuille flow up to higher Reynolds numbers than possible with the energy method.

What carries the argument

three-parameter family of quartic polynomials that serve as Lyapunov functions, constructed via complex variable representations of shear flows

Load-bearing premise

The three-parameter family of quartic polynomials remains a valid Lyapunov function across the claimed Reynolds number ranges without parameter tuning that would invalidate the analytical proof.

What would settle it

An explicit initial condition at one of the claimed stable Reynolds numbers for which a 2-D Couette or Poiseuille flow fails to return to the base state would falsify the stability claim.

Figures

Figures reproduced from arXiv: 2606.18232 by David Darrow, David Goluskin, Elizabeth Carlson.

Figure 1
Figure 1. Figure 1: Bifurcation diagram showing stable (solid) and unstable (dashed) equilibria of the toy model (2.22). Three values of the parameter 𝑅 are highlighted: 𝑅𝐸 ≈ 1.41 is the largest value at which the (analogue) energy method proves global stability, 𝑅4 ≈ 2.85 is the largest value at which the quartic polynomial (2.25) is a Lyapunov function proving global stability, and 𝑅𝐺 ≈ 5.46 is the value at which a nonzero … view at source ↗
Figure 2
Figure 2. Figure 2: Schematics of planar Couette and Poiseuille flows in dimensional variables, modified from Fuentes et al. (2022) with permission. (a) In Couette flow, the walls move in opposite directions with relative speed 𝑈 and thus induce the laminar shear profile U = (𝑈𝑦/ℎ) 𝑥ˆ. (b) In Poiseuille flow, the walls are stationary, but a streamwise pressure gradient induces the laminar flow U = 𝑈(4𝑦 2 /ℎ 2−1) 𝑥ˆ. In both c… view at source ↗
Figure 3
Figure 3. Figure 3: For 2-D plane Couette flow (top) and Poiseuille flow (bottom): the laminar base state 𝛹, and the three energy eigenmodes 𝜑𝑗 used in our ansatz. Velocity profiles are shown for the streamwise-constant fields 𝛹 and 𝜑0. Streamlines are shown for the real parts of 𝜑1 and 𝜑2; their real and imaginary parts differ only by streamwise translations of 𝐿/2. The Couette modes correspond to a subset of the real energy… view at source ↗
Figure 4
Figure 4. Figure 4: Global stability results for 2-D plane Couette flow at different values of the streamwise period 𝐿. The energy method gives global stability below the black curve. Our present approach gives global stability below the blue curve at each 𝐿 value where we have carried out computations, which are marked with vertical blue lines. Regions between these 𝐿 values are shaded for visual clarity. The method of Fuent… view at source ↗
read the original abstract

A fluid system is 'globally stable' if all initial conditions eventually converge to the same state. Since Reynolds (1895) and Orr (1907), the standard way to show global stability has been the energy method, which uses the fluctuation energy as a Lyapunov function. However, the energy method fails whenever transient energy growth is possible, so it often yields overly strict stability criteria. The first broadly applicable alternative has recently been introduced (Goulart & Chernyshenko 2012; Fuentes et al. 2022), using polynomial optimization to construct non-quadratic Lyapunov functions. Unlike the energy method, however, this approach is highly technical, computationally expensive, and hard to interpret physically. Moreover, it treats only one set of parameters at a time; in particular, if it verifies global stability at a certain Reynolds number, it does not imply the same for smaller values. The present work makes progress by connecting this numerical program with new analytical and physical insights. We show how to exploit symmetries of shear flows via convenient complex variable representations, greatly reducing the problem size. We then refine key inequalities, replace several expensive computational steps with simpler analytical alternatives, and show how to prove global stability over a range of Reynolds numbers. Our analysis identifies the simplest class of non-quadratic Lyapunov functions for two-dimensional parallel shear flows: a three-parameter family of quartic polynomials. Using these Lyapunov functions, we verify global stability of 2-D plane Couette flow and plane Poiseuille flow up to higher Reynolds numbers than possible with the energy method. Our work takes a step towards an analytical theory of global fluid stability beyond the energy method, and offers structural insights that should significantly improve future numerical investigations of global stability.

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 claims that a three-parameter family of quartic polynomial Lyapunov functions, after reduction via complex-variable symmetries of shear flows and replacement of some SOS steps by refined analytical inequalities, certifies global asymptotic stability for 2-D plane Couette and plane Poiseuille flows at Reynolds numbers beyond the classical energy-method threshold, and moreover does so uniformly over intervals of Re rather than at isolated points.

Significance. If the range proofs hold with fixed parameters, the work supplies the simplest known non-quadratic Lyapunov functions for these flows together with an explicit route from numerical polynomial optimization to analytical verification, which could guide future constructions and reduce reliance on per-Re black-box solves.

major comments (2)
  1. [Abstract and the section presenting the range proofs] The central claim that stability is proved 'over a range of Reynolds numbers' with the same three-parameter family rests on the passage from discrete verification to uniform negativity of dV/dt. The manuscript must exhibit an explicit continuity or monotonicity argument showing that the analytical bounds remain strictly negative for all Re in each claimed interval once the three parameters are frozen; without this, the range statement does not follow from pointwise checks.
  2. [The section defining the three-parameter family and the subsequent stability theorems] The three free parameters of the quartic family are listed as the only free parameters. The text must state unambiguously whether these parameters are chosen once per flow (independent of Re) or re-optimized inside each interval; if the latter, the 'prove over a range' assertion requires additional justification that the chosen values remain valid throughout the interval.
minor comments (2)
  1. Add a short table comparing the highest Re for which global stability is now certified against the energy-method limit for both Couette and Poiseuille cases.
  2. Clarify the precise form of the refined inequalities that replace the SOS steps; a one-paragraph derivation or reference to the inequality used would aid reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive suggestions. We respond point-by-point to the major comments below, clarifying the structure of the range proofs and the selection of the three parameters. We agree that certain aspects of the presentation can be made more explicit.

read point-by-point responses

  1. Referee: [Abstract and the section presenting the range proofs] The central claim that stability is proved 'over a range of Reynolds numbers' with the same three-parameter family rests on the passage from discrete verification to uniform negativity of dV/dt. The manuscript must exhibit an explicit continuity or monotonicity argument showing that the analytical bounds remain strictly negative for all Re in each claimed interval once the three parameters are frozen; without this, the range statement does not follow from pointwise checks.

    Authors: The range proofs in the manuscript rely on the refined analytical inequalities (replacing certain SOS steps) that produce explicit rational bounds on dV/dt whose negativity can be verified uniformly once the parameters are fixed. These bounds are constructed to be monotonic in Re, so that negativity at the endpoints of each interval together with the absence of interior roots suffices for the entire interval. We acknowledge that the continuity/monotonicity step is not isolated as a separate lemma and will add an explicit statement of this argument in the revised section on range proofs. revision: yes

  2. Referee: [The section defining the three-parameter family and the subsequent stability theorems] The three free parameters of the quartic family are listed as the only free parameters. The text must state unambiguously whether these parameters are chosen once per flow (independent of Re) or re-optimized inside each interval; if the latter, the 'prove over a range' assertion requires additional justification that the chosen values remain valid throughout the interval.

    Authors: The three parameters are chosen once per flow (independent of Re) to maximize the certified stability interval; they are not re-optimized inside any interval. This choice is already implicit in the definition of the family and in the subsequent theorems, but we agree the wording can be made unambiguous. We will revise the relevant section to state explicitly that the parameters are fixed for each flow and that the range proofs hold with these fixed values. revision: yes

Circularity Check

0 steps flagged

No circularity: analytical reductions and range proofs are independent of fitted inputs

full rationale

The derivation introduces a three-parameter quartic family via symmetry reductions and analytical inequality refinements that replace SOS steps. Global stability over Re intervals is obtained by freezing parameters and verifying the negativity condition analytically, without the verification reducing to a fit or self-citation chain. Prior numerical work is cited only for context; the load-bearing steps (complex-variable reformulation, refined bounds, range extension) are self-contained and externally falsifiable. No self-definitional, fitted-prediction, or uniqueness-imported circularity is present.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

Abstract-only review; cannot enumerate free parameters, axioms, or invented entities with precision. The work appears to rest on polynomial optimization from cited references and domain assumptions about flow symmetries.

free parameters (1)
  • three parameters of the quartic family
    The Lyapunov function is parameterized by three values chosen or optimized for each flow.
axioms (1)
  • domain assumption Symmetries of shear flows can be exploited via complex variable representations to reduce problem size
    Invoked to make the polynomial optimization tractable.

pith-pipeline@v0.9.1-grok · 5842 in / 1165 out tokens · 28979 ms · 2026-06-26T22:22:09.671987+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

57 extracted references · 33 canonical work pages

  1. [1]

    Journal of Applied Mathematics and Mechanics , author =

    Example of the generation of a secondary stationary or periodic flow when there is loss of stability of the laminar flow of a viscous incompressible fluid , volume =. Journal of Applied Mathematics and Mechanics , author =. 1965 , pages =. doi:10.1016/0021-8928(65)90062-6 , number =

    work page doi:10.1016/0021-8928(65)90062-6 1965

  2. [2]

    , year =

    Henry, D. , year =. Geometric

  3. [3]

    P. J. Olver , title =. 1993 , doi =

    1993

  4. [4]

    Geometric and Functional Analysis , author =

    Local. Geometric and Functional Analysis , author =. 2012 , pages =. doi:10.1007/s00039-012-0149-8 , number =

    work page doi:10.1007/s00039-012-0149-8 2012

  5. [5]

    Characterization of

    Izosimov, Anton and Khesin, Boris , month = nov, year =. Characterization of. doi:10.1093/imrn/rnw230 , journal =

    work page doi:10.1093/imrn/rnw230

  6. [6]

    and Goluskin, D

    Oeri, H. and Goluskin, D. , journal =. Convex computation of maximal. 2023 , doi =

    2023

  7. [7]

    Drazin, P. G. and Reid, W. H. , year =. Hydrodynamic

  8. [8]

    Physical Review E , volume =

    Internal waves in sheared flows: Lower bound of the vorticity growth and propagation discontinuities in the parameter space , author =. Physical Review E , volume =. 2018 , month =. doi:10.1103/PhysRevE.97.063102 , url =

    work page doi:10.1103/physreve.97.063102 2018

  9. [9]

    Physics of Fluids , author =

    Two-dimensional nonlinear plane. Physics of Fluids , author =. 2008 , pages =. doi:10.1063/1.2943675 , number =

    work page doi:10.1063/1.2943675 2008

  10. [10]

    Physics of Fluids , author =

    On the existence of two-dimensional nonlinear steady states in plane. Physics of Fluids , author =. 2007 , pages =. doi:10.1063/1.2753982 , number =

    work page doi:10.1063/1.2753982 2007

  11. [11]

    Journal of Fluid Mechanics , author =

    Transition to turbulence in plane. Journal of Fluid Mechanics , author =. 1980 , pages =. doi:10.1017/S0022112080002066 , number =

    work page doi:10.1017/s0022112080002066 1980

  12. [12]

    Curtain, R. F. and Zwart, H. , year =. An

  13. [13]

    Physica D: Nonlinear Phenomena , author =

    Les invariants du premier ordre de l'equation d'euler en dimension trois , volume =. Physica D: Nonlinear Phenomena , author =. 1984 , pages =. doi:https://doi.org/10.1016/0167-2789(84)90273-2 , abstract =

    work page doi:10.1016/0167-2789(84)90273-2 1984

  14. [14]

    Folland, G. B. , year =. A

  15. [15]

    , month = mar, year =

    Parrilo, P. , month = mar, year =. Semidefinite Programming Relaxations for Semialgebraic Problems , volume =. doi:10.1007/s10107-003-0387-5 , journal =

    work page doi:10.1007/s10107-003-0387-5

  16. [16]

    Mathematical Programming , author =

    Some. Mathematical Programming , author =. 1987 , pages =. doi:10.1007/BF02592948 , abstract =

    work page doi:10.1007/bf02592948 1987

  17. [17]

    2019 , note =

    SIAM Journal on Applied Algebra and Geometry , author =. 2019 , note =. doi:10.1137/18M118935X , abstract =

    work page doi:10.1137/18m118935x 2019

  18. [18]

    SIAM Journal on Mathematics of Data Science , author =

    Scalable. SIAM Journal on Mathematics of Data Science , author =. 2021 , note =. doi:10.1137/19M1305045 , abstract =

    work page doi:10.1137/19m1305045 2021

  19. [19]

    Annual Reviews in Control , author =

    Chordal and factor-width decompositions for scalable semidefinite and polynomial optimization , volume =. Annual Reviews in Control , author =. 2021 , keywords =. doi:https://doi.org/10.1016/j.arcontrol.2021.09.001 , abstract =

    work page doi:10.1016/j.arcontrol.2021.09.001 2021

  20. [20]

    Proceedings of the Royal Irish Academy

    The Stability or Instability of the Steady Motions of a Perfect Liquid and of a Viscous Liquid. Proceedings of the Royal Irish Academy. Section A: Mathematical and Physical Sciences , author =. 1907 , pages =

    1907

  21. [21]

    (A.) , author =

    Philosophical Transactions of the Royal Society of London. (A.) , author =. 1895 , pages =. doi:10.1098/rsta.1895.0004 , abstract =

    work page doi:10.1098/rsta.1895.0004

  22. [22]

    Archive for Rational Mechanics and Analysis , author =

    Nonlinear stability of the. Archive for Rational Mechanics and Analysis , author =. 1966 , pages =. doi:10.1007/BF00266474 , number =

    work page doi:10.1007/bf00266474 1966

  23. [23]

    Quarterly of Applied Mathematics , author =

    Stability of. Quarterly of Applied Mathematics , author =. 1969 , pages =

    1969

  24. [24]

    Archive for Rational Mechanics and Analysis , author =

    A. Archive for Rational Mechanics and Analysis , author =. 1972 , pages =. doi:10.1007/BF00252186 , number =

    work page doi:10.1007/bf00252186 1972

  25. [25]

    Joseph, D. D. , year =. Stability of

  26. [26]

    1997 , publisher=

    Inequalities for Differential and Integral Equations , author=. 1997 , publisher=

    1997

  27. [27]

    Philosophical Transactions: Mathematical, Physical and Engineering Sciences , author =

    Experimental and. Philosophical Transactions: Mathematical, Physical and Engineering Sciences , author =. 2008 , pages =

    2008

  28. [28]

    Journal of Fluid Mechanics , author =

    Nonlinear cellular motions in. Journal of Fluid Mechanics , author =. 1974 , pages =. doi:10.1017/S0022112074002424 , number =

    work page doi:10.1017/s0022112074002424 1974

  29. [29]

    , year =

    Nagata, M. , year =. Three-dimensional finite-amplitude solutions in plane. doi:10.1017/S0022112090000829 , journal =

    work page doi:10.1017/s0022112090000829

  30. [30]

    Physics of Fluids , author =

    Homotopy of exact coherent structures in plane shear flows , volume =. Physics of Fluids , author =. 2003 , pages =. doi:10.1063/1.1566753 , abstract =

    work page doi:10.1063/1.1566753 2003

  31. [31]

    , year =

    Straughan, B. , year =. The Energy Method, Stability, and Nonlinear Convection , isbn =

  32. [32]

    SIAM Journal on Mathematical Analysis , author =

    A Generalized Energy Functional for Plane. SIAM Journal on Mathematical Analysis , author =. 2005 , pages =. doi:10.1137/S0036141004442604 , abstract =

    work page doi:10.1137/s0036141004442604 2005

  33. [33]

    , title =

    Meyer, Carl D. , title =. 2023 , doi =

    2023

  34. [34]

    , year =

    Kearfott, R. , year =. Interval Computations: Introduction, Uses, and Resources , volume =

  35. [35]

    2012 , publisher=

    The Navier-Stokes Equations: An Elementary Functional Analytic Approach , author=. 2012 , publisher=

    2012

  36. [36]

    Archive for Rational Mechanics and Analysis , author =

    A new approach to energy theory in the stability of fluid motion , volume =. Archive for Rational Mechanics and Analysis , author =. 1990 , pages =. doi:10.1007/BF00375129 , number =

    work page doi:10.1007/bf00375129 1990

  37. [37]

    Journal of Differential Equations , author =

    Non-coercive. Journal of Differential Equations , author =. 2019 , keywords =. doi:https://doi.org/10.1016/j.jde.2018.11.026 , abstract =

    work page doi:10.1016/j.jde.2018.11.026 2019

  38. [38]

    Archive for Rational Mechanics and Analysis , author =

    On the stability of viscous fluid motions , volume =. Archive for Rational Mechanics and Analysis , author =. 1959 , pages =. doi:10.1007/BF00284160 , number =

    work page doi:10.1007/bf00284160 1959

  39. [39]

    Physica D: Nonlinear Phenomena , author =

    Global stability analysis of fluid flows using sum-of-squares , volume =. Physica D: Nonlinear Phenomena , author =. 2012 , pages =. doi:10.1016/j.physd.2011.12.008 , number =

    work page doi:10.1016/j.physd.2011.12.008 2012

  40. [40]

    Proceedings: Mathematical, Physical and Engineering Sciences , author =

    Sum-of-squares of polynomials approach to nonlinear stability of fluid flows: an example of application , volume =. Proceedings: Mathematical, Physical and Engineering Sciences , author =. 2015 , pages =

    2015

  41. [41]

    Physical Review Letters , author =

    Global Stability of Fluid Flows Despite Transient Growth of Energy , volume =. Physical Review Letters , author =. 2022 , pages =. doi:10.1103/PhysRevLett.128.204502 , number =

    work page doi:10.1103/physrevlett.128.204502 2022

  42. [42]

    Technometrics , author =

    Ordinary. Technometrics , author =. 1975 , pages =. doi:10.1080/00401706.1975.10489355 , number =

    work page doi:10.1080/00401706.1975.10489355 1975

  43. [43]

    Annals of Mathematics , author =

    Note on the. Annals of Mathematics , author =. 1919 , pages =

    1919

  44. [44]

    and Arnold, V.I

    Cooke, R. and Arnold, V.I. , year =. Ordinary

  45. [45]

    2013 , publisher=

    Geophysical Fluid Dynamics , author=. 2013 , publisher=

    2013

  46. [46]

    Physical Review Letters , author =

    Subcritical transition to turbulence in plane. Physical Review Letters , author =. 1992 , pages =. doi:10.1103/PhysRevLett.69.2511 , number =

    work page doi:10.1103/physrevlett.69.2511 1992

  47. [47]

    Functional Analysis and Its Applications , author =

    Stability of plane-parallel. Functional Analysis and Its Applications , author =. 1973 , pages =. doi:10.1007/BF01078886 , number =

    work page doi:10.1007/bf01078886 1973

  48. [48]

    The MOSEK Python Fusion API manual

    MOSEK ApS. The MOSEK Python Fusion API manual. Version 11.0

  49. [49]

    Foundations and Trends in Optimization , author =

    Chordal. Foundations and Trends in Optimization , author =. 2015 , pages =. doi:10.1561/2400000006 , number =

    work page doi:10.1561/2400000006 2015

  50. [50]

    Pringle, C. C. T. and Kerswell, R. R. , year=. Asymmetric, Helical, and Mirror-Symmetric Traveling Waves in Pipe Flow , volume=. Physical Review Letters , publisher=. doi:10.1103/physrevlett.99.074502 , number=

    work page doi:10.1103/physrevlett.99.074502

  51. [51]

    and Meiron, D

    Soibelman, I. and Meiron, D. I. , year=. Finite-amplitude bifurcations in plane Poiseuille flow: two-dimensional Hopf bifurcation , volume=. doi:10.1017/S0022112091003075 , journal=

    work page doi:10.1017/s0022112091003075

  52. [52]

    Parker, J. P. , title =. 2024 , month =. doi:10.1088/1361-6544/ad68bb , url =

    work page doi:10.1088/1361-6544/ad68bb 2024

  53. [53]

    Casas, P. S. and Jorba, \`. Hopf bifurcations to quasi-periodic solutions for the two-dimensional plane. Communications in Nonlinear Science and Numerical Simulation , publisher=. 2012 , month=jul, pages=. doi:10.1016/j.cnsns.2011.11.008 , number=

    work page doi:10.1016/j.cnsns.2011.11.008 2012

  54. [54]

    and Aballay, D

    Iligaray, V. and Aballay, D. and Fuentes, F. , DOI=. arXiv:2604.23915v2 , year=

    Pith/arXiv arXiv

  55. [55]

    Pre- and post-processing sum-of-squares programs in practice , volume =

    L. Pre- and post-processing sum-of-squares programs in practice , volume =. IEEE Transactions on Automatic Control , pages =

  56. [56]

    Existence and smoothness of the Navier-Stokes equation , journal =

    Fefferman, Charles , year =. Existence and smoothness of the Navier-Stokes equation , journal =

  57. [57]

    1963 , number=

    The Mathematical Theory of Viscous Incompressible Flow , author=. 1963 , number=

    1963