Pith. sign in

REVIEW 1 major objections 4 minor 54 references

Transition threshold of Couette flow for 2D Boussinesq equations

T0 review · 1 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read This paper proves that 2D Boussinesq Couette flow with Richardson number greater than 1/4 is asymptotically stable against Sobolev perturbations of size at most cκ^(1/3), establishing the transition threshold α = 1/3.

desk verdict Solid extension of the 1/3 threshold to unequal diffusivities at low regularity; the flagged Lemma 2.1 gap is a typo, not a flaw. read the letter →

arxiv 2506.03679 v1 pith:A7QJ6IPT submitted 2025-06-04 math.AP

Xiaoxia Ren , Wei Dongyi This is my paper
classification math.AP MSC 35Q3576E0576D05
keywords CouetteflowBoussinesqequationstransitionthresholdinvisciddampingenhanceddissipationRichardsonnumberSobolevstabilitystratifiedshear
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper establishes the transition threshold α = 1/3 for the 2D Boussinesq equations near Couette flow: if the initial perturbation has Sobolev size at most c(min{ν,μ})^(1/3), the flow stays global and returns to the laminar profile. This matches the optimal threshold known for the unstratified 2D Navier-Stokes equations, so the buoyancy coupling does not worsen the stability window when the Richardson number γ² > 1/4. The proof works for different viscosity ν and thermal diffusivity μ, provided their ratio satisfies (ν+μ)/(2γ√(νμ)) < 2−ε, and it requires only H^(s+1/2) initial data with s > 3/2. A sympathetic reader should care because this quantifies the maximal disturbance a stratified shear flow can absorb before transition, the modern PDE version of Reynolds' classical question.

What carries the argument

The argument is carried by a frequency-adapted energy D(t) = ‖mû‖²_{L²} + γ²‖mϑ̂‖²_{L²} + Re∫(mϑ̂)(mû₁)dξ, which is positive exactly when γ > 1/2. The multiplier m(t,k,ξ) has two regimes: for t ≤ κ^(−1/6) it is |k₊, ξ−kt|^(1/2)⟨k,ξ⟩^s $e^{{M₀}}$, capturing inviscid damping; for t ≥ κ^(−1/6) it adds the factor $e^{{ϵκ^(1/3)t}}$ together with compound terms M₁+M₂+M₃ that suppress echo cascades and capture enhanced dissipation. The cross term Re(mϑ̂ mû₁) combined with the parameter condition (ν+μ)/(2γ√(νμ)) < 2−ε absorbs the linear buoyancy-velocity coupling into the diffusion terms, and symmetrized multiplier bounds reduce all nonlinear terms to cubes of the weighted energy, so that a Grönwall estimate closes on the short time scale and a bootstrap closes on the long time scale.

What would settle it

Compute the linearized Boussinesq evolution with ν = μ = 1 and γ = 1/2, the excluded case where the absorption coefficient in (4.13) equals 2; if the solution still exhibits the exp(−ϵκ^(1/3)t) decay of (1.7), the paper's central claim would be undercut, since the proof's key bound fails exactly there. Alternatively, a numerical simulation of the full nonlinear system at perturbation amplitude κ^(1/3) with (ν+μ)/(2γ√(νμ)) ≥ 2−ε and γ² > 1/4 that transitions to turbulence would falsify the threshold claim.

Watch

Extended reading notes

Core claim

Theorem 1.1 states that for κ = min{ν,μ} ∈ (0,1), γ > 1/2, s > 3/2, and (ν+μ)/(2γ√(νμ)) < 2−ε, there are constants c, ϵ depending only on γ, s, ε such that whenever ‖(u_in, ϑ_in)‖_{$H^{{s+1/2}}$} ≤ cκ^(1/3), the perturbation remains globally bounded and satisfies the inviscid damping estimate ‖(u1)≠‖_{L²} + ⟨t⟩‖u2‖_{L²} + ‖ϑ≠‖_{L²} ≤ C⟨t⟩^(−1/2)e^(−ϵκ^(1/3)t)‖(u_in, ϑ_in)‖ and the enhanced dissipation estimate ‖⟨∇_L⟩^(1/2)(u≠, ϑ≠)‖_{H^s} ≤ Ce^(−ϵκ^(1/3)t)‖(u_in,ϑ_in)‖. In other words, perturbations of size up to κ^(1/3) are absorbed by the Couette mixing mechanism, with the non-zero Fourier modes decaying on the enhanced time scale κ^(−1/3) rather than the ordinary viscous time scale κ^(−1). The regularity assumption H^(s+1/2) is argued to be sharp in the sense that even the linearized Euler-Boussinesq problem needs H² data for the corresponding inviscid damping decay.

Load-bearing premise

The proof needs the ratio (ν+μ)/(2γ√(νμ)) to stay strictly below 2; if that ratio equals or exceeds 2, the linear buoyancy-velocity coupling can no longer be absorbed by diffusion, and the energy estimate that closes the argument breaks down.

Editorial extensions

If this is right

  • If the theorem is correct, stratified Couette flow with γ² > 1/4 has the same Sobolev transition threshold α = 1/3 as 2D Navier-Stokes, so buoyancy does not reduce the basin of attraction at fixed viscosity.
  • The non-zero spatial modes of velocity and temperature decay like e^(−ϵκ^(1/3)t), meaning the effective mixing time is O(κ^(−1/3) log(1/κ)), much faster than the diffusive time O(κ^(−1)).
  • The inviscid damping estimate ⟨t⟩‖u₂‖_{L²} ≤ C⟨t⟩^(−1/2)e^(−ϵκ^(1/3)t) implies the vertical velocity collapses algebraically in the Eulerian frame even while the enhanced dissipation kills the fluctuating part.
  • The result allows ν ≠ μ as long as (ν+μ)/(2γ√(νμ)) < 2−ε, removing the equal-diffusivity restriction of the recent H^N threshold result.
  • The proof only needs H^(s+1/2) initial data with s > 3/2, and the paper argues this regularity is sharp for the inviscid damping effect.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The excluded equality case (ν+μ)/(2γ√(νμ)) = 2 is a natural boundary: the ε-dissipation in the energy estimate shrinks to zero there, so the threshold α = 1/3 may fail or need a different mechanism when viscosity and thermal diffusivity are extremely unequal.
  • At γ = 1/2, the cross-term energy becomes degenerate, so the true transition threshold may be different exactly at the Miles-Howard critical Richardson number; a separate analysis of the critical case would test whether α = 1/3 persists there.
  • The same two-time-scale multiplier and bootstrap strategy should transfer to other dissipative fluid systems with stable stratification or magnetic effects, where the analogous threshold for Couette flow is either open or known only under more restrictive parameter assumptions.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

1 major / 4 minor

Summary. The paper proves a transition threshold of α≤1/3 for the 2D Boussinesq equations around Couette flow in T×R, for Richardson number γ^2>1/4 and possibly different viscosity ν and thermal diffusivity μ. The main theorem, Theorem 1.1, asserts that if the initial perturbation in H^{s+1/2}, s>3/2, has norm at most cκ^{1/3} with κ=min{ν,μ}, and if (ν+μ)/(2γ√νμ)<2−ε, then the solution is global and satisfies inviscid damping and enhanced dissipation estimates. The proof is divided into a short-time regime t≤κ^{-1/6} and a long-time regime t≥κ^{-1/6}, using weighted Fourier multipliers A_k and M, a coupled positive energy, and a large amount of multiplier-difference and convolution analysis to control the nonlinear terms.

Significance. If correct, the result improves the previous α=1/2 threshold for Boussinesq–Couette stability to α=1/3, matching the known 2D Navier–Stokes threshold, and it removes the ν=μ restriction from recent work by Knobel while allowing lower Sobolev regularity. The paper is largely self-contained: the short- and long-time energy propositions are stated precisely, the multiplier lemmas are proved in detail, and the nonlinear estimates are reduced to symmetrization and convolution inequalities. However, a central convolution lemma is misstated and its proof is not valid in the range of parameters actually used; this is a load-bearing gap that is repairable but must be fixed before the theorem is fully supported.

major comments (1)
  1. [Section 2 (Lemma 2.1), used in Lemmas 2.2 and 5.3] Lemma 2.1 is stated with the reciprocal ratio: it claims ∫ dη/(|a,η|^{1+λ}|b,z−η|^{1+λ}) ≤ C |ab|^λ/(|a+b|^λ |a+b,z|^{1+λ}). The proof actually yields, after division, the opposite ratio C |a+b|^λ/(|ab|^λ |a+b,z|^{1+λ}). Moreover, the intermediate inequality |a|^λ+|b|^λ ≤ |a+b|^λ used in the proof is false for 0<λ<1, which is exactly the range λ=δ<1/2 needed in Lemma 5.3. Lemma 2.2 uses λ=2 and would work with the corrected ratio after cancellation of |k|^2, whereas with the stated ratio the resulting bound would grow like k^4 and the summation over k would fail. Lemma 5.3 explicitly requires the corrected ratio: the application with a=⟨l⟩, b=|k−l|, λ=δ gives ⟨l⟩^{-δ}(⟨l⟩+|k−l|)^δ / |⟨l⟩+|k−l|, ξ−(k−l)t|^{1+δ}, not what the stated (2.1) provides. Since Lemma 5.3 controls the Υ-term used in the long-time estimates of Proposition 6.1, the proof of Theorem 1.1 is not fully justified as written. The intended bound is elementary and the gap is repairable, but the statement, the proof, and the applications must be corrected consistently.
minor comments (4)
  1. [Remark 1.1] The statement that the regularity assumption 'should be sharp' is a heuristic comparison with a linear lower bound; no instability or optimality proof is supplied. Please rephrase as a consistency remark or provide a precise statement.
  2. [Section 3, beginning] There is a typo 'A_k(t,ξ)∼∼' with a doubled tilde; it should read 'A_k(t,ξ)∼'.
  3. [Theorem 1.1 and Section 4, Eq. (4.13)] The condition (ν+μ)/(2γ√νμ)<2−ε is an additional restriction on the physical parameters, not a consequence of the Sobolev threshold. It is used to absorb the linear cross term I1; the paper should state this role explicitly and discuss, even briefly, how restrictive it is for γ near 1/2.
  4. [Section 2, proof of Proposition 2.3] In the Gronwall step, the denominator is written as 1−C_+(t+t^2)c1κ^{1/3}; since the differential inequality is dE/dt≤C⟨t⟩E^{3/2}, the exact integration produces factors of 1/2 or 1/4. These can be absorbed into C_+, but the displayed formula should be checked for consistency.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation: the Boussinesq threshold proof is self-contained, with self-citations to [40] and [48] serving only as methodological background.

full rationale

The paper does not exhibit any of the circularity patterns. The theorem's threshold alpha=1/3 is proved by a two-stage bootstrap: Proposition 2.3 with the short-time energy estimate in Proposition 4.1, and Proposition 2.4 with the long-time multiplier estimate in Proposition 6.1. The multipliers A_k and M are explicitly constructed in the paper, not imported as a black box. The only self-references are [40] (Ren-Wei) for the energy structure and [48] (Wei-Zhang) for the two-scale multiplier idea; the relevant estimates are reproved in Sections 4 and 6, and neither citation supplies the Boussinesq threshold conclusion or forbids alternative approaches. The ratio condition (nu+mu)/(2 gamma sqrt(nu mu)) < 2-epsilon is an explicit assumption used to absorb the linear coupling term I1 in (4.13); it is not fitted from the conclusion. No parameter is fitted to data, no known result is renamed, and no uniqueness theorem is invoked. The possible misstatement of Lemma 2.1 (the claimed convolution bound (2.1) does not follow by the displayed division, and Lemma 5.3 uses a reversed-ratio bound) is a correctness risk to be checked, not a circularity: it does not make the derivation equivalent to its inputs. Hence no significant circularity; the score reflects only minor, non-load-bearing self-citation.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central claim rests on the Boussinesq model, the steady Couette state, the stability condition γ>1/2, the ratio condition on ν and μ, and standard mathematical tools. No parameters are fitted to data, and no new physical entities are introduced. The multipliers A_k and M are analytic devices, not invented entities.

assumptions (6)
  • domain assumption 2D Boussinesq equations with viscosity ν and thermal diffusivity μ are the correct model.
    Stated as the starting point in (1.1); the result is a theorem about this system.
  • standard math The background Couette flow v_s=(y,0), ρ_s=-γ^2 y+1 is a steady state.
    Verification of (1.2) is routine; the perturbation coordinates are defined in (1.3).
  • domain assumption Richardson number condition γ^2>1/4 (equivalently γ>1/2) makes the quadratic form D(t) in (1.8) positive definite.
    Used throughout; see the definition of D(t) and the energy equivalences before (2.3).
  • domain assumption The ratio condition (ν+μ)/(2γ√νμ)<2-ε is imposed.
    Used to absorb the linear cross-coupling term I1 in (4.13); this is a genuine restriction on parameters.
  • standard math Standard analytic tools: Fourier transform, Plancherel, Young and Hölder inequalities, Grönwall's lemma.
    These are invoked throughout Sections 2-6 without proof.
  • standard math The convolution inequality in Lemma 2.1 is an elementary inequality and is proven in the paper.
    Proven in Section 2; we list it as a background estimate rather than a free assumption.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Transition threshold of Couette flow for 2D Boussinesq equations." pith.science (2026). https://pith.science/paper/A7QJ6IPT

@misc{pith2026250603679,
  author       = {Pith},
  title        = {Pith review of: Transition threshold of Couette flow for 2D Boussinesq equations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/A7QJ6IPT}},
  note         = {Machine review of arXiv:2506.03679}
}
abstract

In this paper, we prove the stability threshold of $\alpha\leq \frac13$ for 2D Boussinesq equations around the Couette flow in $\mathbb{T}\times \mathbb{R}$ with Richardson number $\gamma^2>\frac14$ and different viscosity $\nu$ and thermal diffusivity $\mu$. More precisely, if $\|v_{in}-(y,0)\|_{H^{s+1/2}}+ \|\rho_{in}+\gamma^2 y-1\|_{H^{s+1/2}}\leq c(\min\{\nu,\mu\})^{1/3}$, $\frac{\nu+\mu}{2\gamma\sqrt{\nu \mu} }< 2-\varepsilon$, $s>3/2$, then the asymptotic stability holds. This stability threshold is consistent with the optimal stability threshold for the 2D Navier-Stokes equations in Sobolev space. And in the sense of inviscid damping effect, the regularity assumption of the initial data should be sharp.

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

54 extracted references · 54 canonical work pages

  1. [29]

    Suppression of Fluid Echoes and Sobolev Stability Threshold for 2D Dissipative Fluid Equations Around Couette Flow

    N. Knobel, Suppression of fluid echoes and Sobolev stability threshold for 2D dissipative fluid equations around Couette flow, arXiv:2505.23391v1

    work page Pith review arXiv

  2. [1]

    Enhanced Dissipation, Taylor Dispersion, and Inviscid Damping of Couette flow in the Boussinesq system on the Plane

    R. Arbon, Enhanced Dissipation, Taylor Dispersion, and Inviscid Damping of Couette flow in the Boussi- nesq system on the Plane, 2025, arXiv:2501.00690

    work page Pith review arXiv 2025

  3. [2]

    Bedrossian, R

    J. Bedrossian, R. Bianchini, M. Coti Zelati, and M. Dolce, Nonlinear inviscid damping and shear-buoyancy instability in the two-dimensional Boussinesq equations, Comm. Pure Appl. Math. 76(12), 2023, 3685- 3768

    work page 2023

  4. [3]

    Bedrossian, P

    J. Bedrossian, P. Germain and N. Masmoudi, On the stability threshold for the 3D Couette flow in Sobolev regularity, Ann. of Math., 185, 2017, no. 2, 541-608

    work page 2017

  5. [4]

    Bedrossian, P

    J. Bedrossian, P. Germain and N. Masmoudi, Stability of the Couette flow at high Reynolds number in 2D and 3D, Bull. Amer. Math. Soc., 56, 2019, 373-414

    work page 2019

  6. [5]

    Bedrossian, P

    J. Bedrossian, P. Germain and N. Masmoudi, Dynamics near the subcritical transition of the 3D Couette flow I: below threshold case, Mem. Am. Math. Soc., 266, 2020, v+158

    work page 2020

  7. [6]

    Bedrossian and N

    J. Bedrossian and N. Masmoudi, Inviscid damping and the asymptotic stability of planar shear flow in the 2D Euler equations, Publ. Math. Inst. Hautes ´Etudes Sci., 122, 2015, 195-300

    work page 2015

  8. [7]

    Bedrossian, N

    J. Bedrossian, N. Masmoudi and V. Vicol, Enhanced dissipation and inviscid damping in the inviscid limit of the Navier-Stokes equations near the two dimensional Couette flow, Arch. Ration. Mech. Anal., 219, 2016, 1087-1159

    work page 2016

Show all 54 references
  1. [8]

    Bedrossian, V

    J. Bedrossian, V. Vicol and F. Wang, The Sobolev stability threshold for 2D shear flows near Couette, J. Nonlinear Sci., 28, 2018, 2051-2075

    2018

  2. [9]

    Bianchini, M

    R. Bianchini, M. Coti Zelati and M. Dolce, Linear inviscid damping for shear flows near Couette in the 2D stably stratified regime, Indiana Univ. Math. J. 71(4), 2022, 1467-1504

    2022

  3. [10]

    Q. Chen, T. Li, D. Wei, and Z. Zhang, Transition threshold for the 2-D Couette flow in a finite channel, Arch. Ration. Mech. Anal., 238, 2020, 125-183

    2020

  4. [11]

    Q. Chen, D. Wei, and Z. Zhang, Transition threshold for the 3-D Couette flow in a finite channel, Mem. Amer. Math. Soc., 296, 2024, no. 1478, v+178 pp

    2024

  5. [12]

    Chimonas, Algebraic disturbances in stratified shear flows, J

    G. Chimonas, Algebraic disturbances in stratified shear flows, J. Fluid Mech., 90, 1999, 1-19

    1999

  6. [13]

    Coti Zelati, T

    M. Coti Zelati, T. M. Elgindi, and K. Widmayer, Enhanced dissipation in the Navier-Stokes equations near the Poiseuille flow, Comm. Math. Phys., 378, 2020, 987-1010. TRANSITION THRESHOLD FOR 2D BOUSSINESQ EQUATIONS 31

    2020

  7. [14]

    Coti Zelati and M

    M. Coti Zelati and M. Nualart, Explicit solutions and linear inviscid damping in the Euler-Boussinesq equation near a stratified Couette flow in the periodic strip, J. Hyperbol. Differ. Eq., 2025, doi: 10.1142/S0219891624400058

    2025 doi

  8. [15]

    Coti Zelati, A

    M. Coti Zelati, A. Zotto and K. Widmayer, Stability of viscous three-dimensional stratified Couette flow via dispersion and mixing, 2024, arXiv:2402.15312

    2024 arXiv

  9. [16]

    Daviaud, J

    F. Daviaud, J. Hagseth and P. Berg´ e, Subcritical transition to turbulence in plane Couette flow, Phys. Rev. Lett., 69, 1992, 2511-2514

    1992

  10. [17]

    W. Deng, J. Wu and P. Zhang, Stability of Couette flow for 2D Boussinesq system with vertical dissipation, J. Funct. Anal., 281(12), 2021, 109255

    2021

  11. [18]

    Ding and Z

    S. Ding and Z. Lin, Enhanced dissipation and transition threshold for 2-D plane Poiseuille flow via resolvent estimate, J. Differ Equ., 332, 2022, 404-439

    2022

  12. [19]

    P. G. Drazin, The stability of a shear layer in an unbounded heterogeneous inviscid fluid, J. Fluid Mech, 4, 1958, 214-224

    1958

  13. [20]

    Drazin and W

    P. Drazin and W. Reid, Hydrodynamic Stability, Cambridge Monographs Mech. Appl. Math., Cambridge Univ. Press, New York, 1981

    1981

  14. [21]

    B. F. Farrell and P. J. Ioannou, Transient development of perturbations in stratified shear flow, J. Atmos. Sci., 50, 1993, 2201-2214

    1993

  15. [22]

    A. E. Gill, Atmosphere-ocean dynamics. Academic press, 1982

    1982

  16. [23]

    L. N. Howard, Note on a paper of John W. Miles, J. Fluid Mech., 10(4), 1961, 509-512

    1961

  17. [24]

    Ibrahim, Y

    S. Ibrahim, Y. Maekawa, and N. Masmoudi, On pseudospectral bound for non-selfadjoint operators and its application to stability of Kolmogorov flows, Ann. PDE, 5(2), 2019, 1-84

    2019

  18. [25]

    A. D. Ionescu and H. Jia, Nonlinear inviscid damping near monotonic shear flows, Acta Math. 230(2), 2023, 321-399

    2023

  19. [26]

    A. D. Ionescu and H. Jia. Inviscid damping near the Couette flow in a channel, Comm. Math. Phys., 374(3), 2020, 2015-2096

    2020

  20. [27]

    J. Jin, X. Ren and D. Wei, Stability threshold of Couette flow for 2D MHD equations in the Sobolev space, Nonlinearity, 2025, doi: 10.1088/1361-6544/adcb81

    2025 doi

  21. [28]

    Kelvin, Stability of fluid motion-rectilinear motion of viscous fluid between two parallel plates, Phil

    L. Kelvin, Stability of fluid motion-rectilinear motion of viscous fluid between two parallel plates, Phil. Mag. 24, 1887, 188-196

  22. [30]

    H. Li, N. Masmoudi and W. Zhao, A dynamical approach to the study of instability near Couette flow, Comm. Pure Appl. Math., 77(6), 2024, 2863-2946

    2024

  23. [31]

    T. Li, D. Wei and Z. Zhang, Pseudospectral bound and transition threshold for the 3D Kolmogorov flow, Comm. Pure Appl. Math., 73, 2020, 465-557

    2020

  24. [32]

    M. Lai, R. Pan and K. Zhao, Initial boundary value problem for two-dimensional viscous Boussinesq equations, Arch. Ration. Mech. Anal., 199(3), 2011: 739-760

    2011

  25. [33]

    Majda, Introduction to PDEs and Waves for the Atmosphere and Ocean

    A. Majda, Introduction to PDEs and Waves for the Atmosphere and Ocean. IEEE, 2003

    2003

  26. [34]

    Masmoudi, B

    N. Masmoudi, B. Said-Houari, and W. Zhao, Stability of Couette flow for 2D Boussinesq system without thermal diffusivity, Arch. Ration. Mech. Anal. 245(2), 2022, 645-752

    2022

  27. [35]

    Masmoudi, C

    N. Masmoudi, C. Zhai and W. Zhao, Asymptotic stability for two-dimensional Boussinesq systems around the Couette flow in a finite channel, J. Funct. Anal., 284(1), 2023, 109736

    2023

  28. [36]

    Masmoudi and W

    N. Masmoudi and W. Zhao, Nonlinear inviscid damping for a class of monotone shear flows in finite channel, Ann. of Math. (2), 199(3), 2024, 1093-1175

    2024

  29. [37]

    Masmoudi and W

    N. Masmoudi and W. Zhao, Stability threshold of two-dimensional Couette flow in Sobolev spaces, Ann. Inst. Henri Poincar´ e, 39, 2022, 245-325

    2022

  30. [38]

    J. W. Miles, On the stability of heterogeneous shear flows, J. Fluid Mech., 10, 1961, 496-508

    1961

  31. [39]

    Niu and W

    B. Niu and W. Zhao, Improved stability threshold of the Two-Dimensional Couette flow for Navier-Stokes- Boussinesq Systems via quasi-linearization, Commun. Pur. Appl. Anal., 2025, doi:10.3934/cpaa.2025029

    2025 doi

  32. [40]

    Ren and D

    X. Ren and D. Wei, On the stability threshold of Couette flow for 2D Boussinesq equations, Nonlinear Anal.Real., Accepted

  33. [41]

    Pedlosky, Geophysical Fluid Dynamics, Second Edition

    J. Pedlosky, Geophysical Fluid Dynamics, Second Edition. 1987

    1987

  34. [42]

    O. Reynolds, An experimental investigation of the circumstances which determine whether the motion of water shall be direct or sinuous, and of the law of resistance in parallel channels, Philos. Trans. R. Soc. Lond. 174, 1883, 935-982

  35. [43]

    V. A. Romanov, Stability of plane-parallel Couette flow, Funkcional. Anal. i Prilozen, 7, 1973, 62-73

    1973

  36. [44]

    Tillmark and P

    N. Tillmark and P. H. Alfredsson, Experiments on transition in plane Couette flow, J. Fluid Mech., 235, 1992, 89-102. 32 XIAOXIA REN AND DONGYI WEI

    1992

  37. [45]

    Trefethen and A

    L. Trefethen and A. Trefethen, S. Reddy and T. Driscoll, Hydrodynamic stability without eigenvalues. Science, 261, 1993, 578-584

    1993

  38. [46]

    G. I. Taylor, Effect of variation in density on the stability of superposed streams of fluid, Proc. Royal Society London. A., 132 (820), 1931, 499-523

    1931

  39. [47]

    Wei and Z

    D. Wei and Z. Zhang, Transition threshold for the 3D Couette flow in Sobolev space, Comm. Pure Appl. Math., 74 (11), 2020, 2398-2479

    2020

  40. [48]

    Wei and Z

    D. Wei and Z. Zhang, Nonlinear enhanced dissipation and inviscid damping for the 2D couette flow, Tunis. J. Math., 5, 2023, 573–592

    2023

  41. [49]

    D. Wei, Z. Zhang, and W. Zhao, Linear inviscid damping and enhanced dissipation for the Kolmogorov flow, Adv. Math., 362, 2020, 106963, 103

    2020

  42. [50]

    Wei and Z

    D. Wei and Z. Zhang, Stability threshold of the 2D Couette flow in a finite channel, preprint

  43. [51]

    Yang and Z

    J. Yang and Z. Lin, Linear inviscid damping for Couette flow in stratified fluid, J. Math. Fluid Mech., 20, 2018, 445-472

    2018

  44. [52]

    Zhang and R

    Z. Zhang and R. Zi. Stability threshold of Couette flow for 2D Boussinesq equations in Sobolev spaces, J. Math. Pures Appl., 179, 2023, 123-182

    2023

  45. [53]

    Zhai and W

    C. Zhai and W. Zhao. Stability threshold of the Couette flow for Navier-Stokes Boussinesq system with large Richardson numberγ 2 > 1 4 . SIAM J. Math. Anal., 55, 2023, 1284-1318

    2023

  46. [54]

    Zillinger, On enhanced dissipation for the Boussinesq equations, J

    C. Zillinger, On enhanced dissipation for the Boussinesq equations, J. Differ Equ., 282, 2021, 407-445. Department of Mathematics and Physics, North China Electric Power University, 102206, Beijing, P. R. China Email address:xiaoxiaren@ncepu.edu.cn School of Mathematical Scien...

    2021

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.