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The stability threshold for 3D MHD equations around Couette with rationally aligned magnetic field

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper proves that for 3D magnetohydrodynamic flow near Couette shear, a rationally aligned magnetic field still yields the optimal stability threshold gamma=1, with nonzero magnetic modes amplifying at most like nu^{-1/3}.

desk verdict Proves γ=1 for rational σ only under a strong-field condition |α|>8p that the abstract omits; the resonant-mode analysis is genuinely new and worth reviewing. read the letter →

arxiv 2505.19822 v3 pith:CUF4FT2J submitted 2025-05-26 math.AP

Fei Wang , Lingda Xu , Zeren Zhang This is my paper · ORCID
classification math.AP MSC 35Q3576E2576W05
keywords stabilitythreshold3DMHDequationsCouetteflowrationalmagneticfieldenhanceddissipationinvisciddampingSobolevspacesFouriermultipliermethod
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

The paper proves that for the 3D incompressible MHD equations near Couette flow, when the background magnetic field is rationally aligned, the stability threshold is gamma=1: initial perturbations of size epsilon_0 nu remain globally controlled, with velocity fluctuations decaying through enhanced dissipation at rate $nu^{{1/3}}$ and the magnetic field's nonzero modes amplifying by at most $nu^{{-1/3}}$. This covers all rational slopes $\sigma$=q/p under the strong-field condition |$\alpha$|>8p, and it answers an open question raised in earlier work for generically irrational fields. The proof's central move is to separate Fourier modes where the rational alignment makes the background magnetic term vanish exactly (homogeneous modes) from all other modes, and to show that a loss of one derivative of regularity suffices to control their interaction. If correct, the same gamma=1 threshold holds for rational and Diophantine-irrational alignments alike, in contrast to the weaker gamma=4/3 known for arbitrary $\sigma$.

What carries the argument

The proof is carried by two mechanisms working together. First, the rational slope $\sigma$=q/p forces the resonance set $\sigma$ k + l = 0 to be a genuine sublattice, so every nonzero mode splits into 'homogeneous' modes ($\sigma$ k + l = 0, where the background field exerts no restoring force) and 'non-homogeneous' modes (|$\sigma$ k + l| >= 1/p, where the oscillatory multiplier T^t_{\pm\$\alpha$}=$e^{{\mp i\alpha(\sigma k+l)t}}$ can be integrated by parts). Second, the good unknowns $W^{{\pm}}$=$T^{{\pm}}$(U\pm B) and the vorticity-type variables Q=\Delta_L U, G=\Delta_L B reveal that the linearized homogeneous modes decouple: $Q^{2}$_{\neq H} obeys a purely damped equation, while $G^{2}$_{\neq H} picks up the stretching term 2\partial^L_{XY}\$Delta_L^{{-1}}$$G^{2}$_{\neq H}, producing the \$nu^{{-2/3}}$ linear amplification that becomes \$nu^{{-1/3}}$ after nonlinear coupling. The Fourier multiplier M=$e^{{\delta_0\nu^{1/3}}$t}M_1M_2 (and M_3 for the zero mode) converts these mechanisms into the exponential decay rate \delta_0\$nu^{{1/3}}$ and the \$nu^{{-1/6}}$ $L^{2}$-time growth seen in the estimates.

What would settle it

Run the Fourier-truncated 3D MHD system at a rational $\sigma$=q/p with $\alpha$=9p (inside the theorem's range) and initial data of size epsilon_0 nu; if any nonzero magnetic mode grows faster than C $nu^{{-1/3}}$ or the nonzero velocity modes fail to show the $nu^{{1/3}}$ enhanced-dissipation rate for arbitrarily small epsilon_0, the threshold claim fails. A cheaper consistency check is the boundary constant itself: in the energy estimate (3.1) the oscillatory term is bounded by p/(2|$\alpha$|) times the bootstrap quantities, and |$\alpha$|>8p is exactly the margin that makes that factor harmless.

Watch

Extended reading notes

Core claim

The paper's central claim is Theorem 1.1: with nu=mu, rational $\sigma$=q/p, |$\alpha$|>8p, and N>9/2, any initial datum satisfying ||(u_in,b_in)||_{$H^{{N+2}}$} <= epsilon_0 nu leads to a global solution whose sheared-coordinate profiles satisfy the decay estimates (1.3a)-(1.3f). Concretely, the nonzero velocity modes lose energy at the enhanced rate delta_0 $nu^{{1/3}}$, the $U^{2}$_neq component exhibits nonlinear inviscid damping, the nonzero magnetic modes are amplified by at most $nu^{{-1/3}}$ (with partial_X $B^{1}$_neq reaching $nu^{{-1/2}}$), and the zero modes stay uniformly bounded, so the lift-up mechanism is suppressed. The new content is that these bounds hold for rational $\sigma$, where the generic Diophantine condition used in prior work fails; the proof replaces it with the exact spectral gap |$\sigma$ k + l| >= 1/p and isolates the homogeneous modes $\sigma$ k + l = 0 as the only place where the field's restoring force disappears.

Load-bearing premise

The argument assumes the background magnetic field is strong enough, |alpha|>8p for sigma=q/p, and without this inequality the oscillatory estimates in Section 3.1 do not close, so the claimed gamma=1 threshold is only proven in the strong-field regime.

Editorial extensions

If this is right

  • For every rational alignment sigma=q/p with a sufficiently strong field, perturbations of size epsilon_0 nu remain globally stable, matching the gamma=1 threshold already known for Diophantine irrational sigma.
  • The nonzero magnetic field modes grow by at most nu^{-1/3} (and partial_X B^1_neq by nu^{-1/2}), so the magnetic field does not stay small on the enhanced dissipation time scale; this growth is part of the stable picture, not a sign of instability.
  • Velocity modes with nonzero Fourier frequency enjoy inviscid damping: U^2_neq decays uniformly in H^{N-2} while the sheared-gradient contribution has the nu^{1/3} enhanced-dissipation rate; this is the first nonlinear inviscid damping statement for u^2_neq in this rational setting.
  • The zero modes (U^0,B^0) remain bounded in H^N, so the lift-up mechanism that would linearly amplify the streamwise velocity is suppressed at this perturbation size.
  • As the authors note, the same proof strategy with unequal viscosities nu != mu gives a corresponding statement for data of size min{nu,mu} when |alpha| >= 8p(mu+nu)/sqrt(mu nu).

Reading between the lines

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

  • If the gap condition |sigma k + l| >= 1/p is the real organizing mechanism, the same gamma=1 threshold should hold for all rational sigma with |alpha|>8p, and possibly for badly approximable irrationals once a suitable quantitative gap is assumed; this is a testable conjecture, not a claim of the paper.
  • The predicted nu^{-1/3} magnetic amplification at rational sigma could be checked by direct numerical simulation of the linearized system (or of the full MHD equations at small epsilon); seeing a steeper growth of partial_X B^1_neq would indicate a missing nonlinear stabilization channel.
  • The strong-field cutoff |alpha|>8p is set by the proof's integration-by-parts constant; one would expect the threshold to change character near alpha approx 8p, where the boundary term p/(2|alpha|) crosses the bootstrap margin — this boundary regime is left open.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. This paper studies the stability threshold for the 3D incompressible MHD system on T x R x T near Couette flow (y,0,0) with a uniform magnetic field alpha(sigma,0,1), where sigma = q/p is rational and nu = mu. The main theorem (Theorem 1.1) asserts that if |alpha| > 8p and the initial perturbation has H^{N+2} norm epsilon <= epsilon_0 nu with N > 9/2, then global stability holds. The proof establishes enhanced dissipation of order nu^{1/3}, inviscid damping of U^2_neq, a nu^{-1/3} amplification of nonzero-mode magnetic fields, and suppression of the lift-up effect for the zero mode, encoded in the estimates (1.3a)-(1.3f). The strategy splits frequencies into homogeneous and non-homogeneous modes with respect to |sigma k + l|, introduces good unknowns W^+- = T^t_{+-alpha}(U +- B), and closes a bootstrap (Proposition 2.1) using the multipliers M1, M2, M3 and energy estimates for the non-homogeneous, homogeneous, and zero modes. The title and abstract advertise the threshold gamma = 1 for rationally aligned fields, but the theorem itself requires the strong-field condition |alpha| > 8p, a point that is load-bearing in the proof.

Significance. If fully verified, the result would improve the known threshold for rational sigma from gamma = 4/3 to gamma = 1, complementing the Diophantine irrational result of Liss and the recent extension by Rao, Zhang, and Zi, and it would provide a new nu^{-1/3} magnetic amplification in low Sobolev regularity. The paper's conceptual tools are well chosen: separating homogeneous modes sigma k + l = 0, using the M3 multiplier of Wei-Zhang, and carefully trading regularity for growth estimates are natural and potentially useful for later work. The main estimates are stated in explicit, checkable form, and the bootstrap does not fit any parameter to the claimed conclusion, which is a genuine strength. However, the proof as written leaves several load-bearing estimates as sketches, and the advertised scope of the result is broader than what Theorem 1.1 actually proves; these issues must be addressed before the result can be fully evaluated.

major comments (3)
  1. [Sec. 1, Theorem 1.1; Sec. 3.1] The abstract and title claim the gamma = 1 threshold for rationally aligned magnetic fields, but Theorem 1.1 assumes |alpha| > 8p, and Section 3.1 shows this assumption is load-bearing. In the OLS estimate following Eq. (3.1), the boundary and bulk terms give OLS1 + OLS2 <= p/(2|alpha|)(||dX|grad L|M W^2_{neq NH}||^2_{L^infty H^N} + ||dXX M W^2_{neq NH}||^2_{L^2 H^N}), and the proof absorbs them by invoking |alpha| > 8p, which makes p/(2|alpha|) < 1/16. For |alpha| <= 8p the paper supplies no alternative estimate, and the bootstrap hypothesis (2.9a) cannot be closed. The proven statement is therefore a strong-field theorem for each rational sigma, with the required field strength growing with the denominator. Please revise the abstract and title to state this restriction explicitly, and add to Remark 1.2 that for a fixed alpha the result covers only denominators p < |alpha|/8.
  2. [Sec. 2.4, Lemma 2.1] Lemma 2.1, local well-posedness, is stated with the sentence 'we state the following lemma without showing more details' and with no reference. This lemma provides the existence interval [0,2t0] and the initial bounds at t0 that are needed for the continuity argument in Proposition 2.1, so it is part of the proof of the main theorem. Please include a proof, or at least a precise statement of the standard theorem being invoked together with the exact Sobolev regularity and the dependence of t0 on alpha and sigma. As written, the bootstrap has no demonstrated starting point.
  3. [Secs. 3.1, 3.2, 4.2, 4.3, 4.5, 5.1] Several estimates that are load-bearing for Proposition 2.1 are omitted as 'similar' or 'left to the reader'. Examples include the remaining contributions of NLT and NLP in Section 3.1, the other contributions of NLS1, NLS2, and NLP in Section 3.2, the NLS2 and NLT terms in Section 4.2, the nonlinear terms in Sections 4.3 and 4.5, and the LU3 and remaining terms in Section 5.1. Some of these involve the same homogeneous-homogeneous interactions and zero-mode terms that the paper identifies as the main difficulty. To make the proof verifiable, please provide complete estimates in an appendix, or give a precise accounting of why each omitted case reduces to a displayed estimate. The current level of detail is not sufficient for a journal referee to certify that the bootstrap closes at the stated nu-scales.
minor comments (5)
  1. [Abstract; Sec. 1] The abstract states the threshold 'in H^N (N > 13/2)', while Theorem 1.1 states N > 9/2 and measures the initial data in H^{N+2}. These are consistent after reindexing, but the mismatch should be explained explicitly to avoid an apparent contradiction.
  2. [Sec. 1, Theorem 1.1] The theorem states sigma = q/p in Q without requiring gcd(q,p) = 1. Since the condition |alpha| > 8p depends on the chosen representative, please add the reduced-form assumption or state that any denominator p works with the correspondingly weaker bound.
  3. [Sec. 4.4] In the first energy estimate for dY^L M B^2_neq H, the right-hand side displays dX M U^3_neq H in the initial term and in the L^2 H^{N'} term; these should be dY^L M B^2_neq H. Please correct the typo.
  4. [Sec. 4.5] In the energy estimate for dX^s M B^3_neq H, the initial term is written as dX^s M U^3_neq H(t0); it should be dX^s M B^3_neq H(t0).
  5. [Global] There are several typographical errors: 'magnetic fleld' in the abstract, 'magnetic flied' in the introduction, 'Turning to' spelled 'Turing to' in Section 3.3, and 'simliar' in Section 3.4. Please proofread the text.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the γ=1 threshold for rational σ is proved by a bootstrap over explicit energy estimates under the stated strong-field hypothesis |α|>8p; the cited tools are external and not the target claim.

full rationale

The proof is a self-contained bootstrap. Theorem 1.1 assumes ν=μ, σ=q/p, |α|>8p, and ||(u_in,b_in)||_{H^{N+2}}≤ε0ν, and then establishes the estimates (1.3a)-(1.3f) with right-hand sides of order ε or ν^{-1/3}ε. No quantity in the conclusion is fitted from the data, and no estimate is reused as its own input: the bootstrap hypotheses (2.9)-(2.12) are improved by energy estimates that produce strictly smaller constants, which is the standard way of closing an argument, not a circular reduction. The OLS boundary terms in Section 3.1 are absorbed using the spectral gap |σk+l|≥1/p and the hypothesis |α|>8p; this is a stated assumption, not a hidden copy of the conclusion. The oscillation multiplier is attributed to [27] and the multiplier M3 to [33]; these are external tools used as black boxes, and neither is the claim that γ=1 for rational σ. The self-citation [31] concerns the 2D MHD problem with general viscosities and is not load-bearing for the 3D rational-sigma theorem. Lemma 2.1 is a standard local well-posedness statement and is not used to force the result. The main caveat is that the abstract and title advertise the rational-alignment threshold without the |α|>8p condition, while Theorem 1.1 proves only a strong-field version with the field strength growing with the denominator p; this is a scope limitation, not circularity. Therefore no specific reduction of a conclusion to an input can be identified, and the circularity score is 0.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The proof relies on standard Sobolev and Fourier calculus, on prior multiplier estimates from [27,33], and on two structural assumptions of the theorem: rationality of sigma and the strong-field lower bound |alpha|>8p. No new physical entities or fitted parameters are introduced; the free parameters listed are internal proof constants, not data-fitted.

free parameters (3)
  • Bootstrap constant C0 = exists, large (not specified)
    Introduced in Section 2.4 to close the bootstrap; the theorem only requires existence, no numerical value is assigned.
  • Smallness threshold epsilon_0 = epsilon_0(N,sigma) > 0 (not quantified)
    Stated in Theorem 1.1 as depending on N and sigma, but not computed; it is a proof device ensuring the bootstrap starts.
  • Multiplier decay constant delta_0 = 1/(100|alpha|)
    Chosen in Section 2.3 so that the exponential decay rate delta_0 nu^{1/3} is absorbed by the dissipative terms in (3.2); not a physical parameter.
assumptions (4)
  • standard math Local well-posedness of the 3D MHD system in Sobolev spaces
    Lemma 2.1 in Section 2.4 is stated without proof; the bootstrap argument depends on it to obtain a solution on [0,2t_0].
  • domain assumption Spectral gap |sigma k + l| >= 1/p for rational sigma=q/p with k != 0 and sigma k + l != 0
    This gap is a consequence of sigma in Q and is used to decompose modes into non-homogeneous and homogeneous classes (2.1)-(2.2). It is essential in the OLS estimates in Section 3.1.
  • domain assumption Strong-field condition |alpha| > 8p
    Assumed in Theorem 1.1 and used, together with the spectral gap, to control OLS boundary terms in Section 3.1; without it the bootstrap estimates fail.
  • standard math Properties of multipliers M1, M2 from [27] and M3 from [33]
    The energy estimates rely on the decay properties (2.8) and the M3 multiplier bounds (Section 2.3) proved in the cited prior works; they are used as black boxes.

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Pith. "Pith review of The stability threshold for 3D MHD equations around Couette with rationally aligned magnetic field." pith.science (2026). https://pith.science/paper/CUF4FT2J

@misc{pith2026250519822,
  author       = {Pith},
  title        = {Pith review of: The stability threshold for 3D MHD equations around Couette with rationally aligned magnetic field},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CUF4FT2J}},
  note         = {Machine review of arXiv:2505.19822}
}
abstract

We address a stability threshold problem of the Couette flow $(y,0,0)$ in a uniform magnetic fleld $\alpha(\sigma,0,1)$ with $\sigma\in\mathbb{Q}$ for the 3D MHD equations on $\mathbb{T}\times\mathbb{R}\times\mathbb{T}$. Previously, the authors in \cite{L20,RZZ25} obtained the threshold $\gamma=1$ for $\sigma\in\mathbb{R}\backslash\mathbb{Q}$ satisfying a generic Diophantine condition, where they also proved $\gamma = 4/3$ for a general $\sigma\in\mathbb{R}$. In the present paper, we obtain the threshold $\gamma=1$ in $H^N(N>13/2)$, hence improving the above results when $\sigma$ is a rational number. The nonlinear inviscid damping for velocity $u^2_{\neq}$ is also established. Moreover, our result shows that the nonzero modes of magnetic field has an amplification of order $\nu^{-1/3}$ even on low regularity, which is very different from the case considered in \cite{L20,RZZ25}.

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Forward citations

Cited by 1 Pith paper

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Works this paper leans on

37 extracted references · 32 canonical work pages · cited by 1 Pith paper

  1. [27]

    K. Liss. On the sobolev stability threshold of 3d couette flow in a uniform magnetic field. Communications in Mathematical Physics, 377(2):859–908, 2020

    work page 2020

  2. [30]

    Y. Rao, Z. Zhang, and R. Zi. Stability threshold of couette flow for the 3d mhd equations. Journal of Functional Analysis, 288(5):110796, 2025

    work page 2025

  3. [31]

    Wang and Z

    F. Wang and Z. Zhang. The stability threshold for 2d mhd equations around couette with general viscosity and magnetic resistivity.arXiv preprint arXiv:2410.20404, 2024

    arXiv 2024

  4. [1]

    Antonelli, M

    P. Antonelli, M. Dolce, and P. Marcati. Linear stability analysis of the homogeneous couette flow in a 2d isentropic compressible fluid.Annals of PDE, 7(2):24, 2021

    work page 2021

  5. [2]

    Quantitative Hydrodynamic Stability for Couette Flow on Unbounded Domains with Navier Boundary Conditions

    R. Arbon and J. Bedrossian. Quantitative hydrodynamic stability for couette flow on un- bounded domains with navier boundary conditions.arXiv preprint arXiv:2404.02412, 2024

    work page Pith review arXiv 2024

  6. [3]

    Bedrossian, P

    J. Bedrossian, P. Germain, and N. Masmoudi. On the stability threshold for the 3d couette flow in sobolev regularity.Annals of Mathematics, 185(2):541–608, 2017

    work page 2017

  7. [4]

    Bedrossian, P

    J. Bedrossian, P. Germain, and N. Masmoudi. Dynamics near the subcritical transition of the 3D Couette flow I: Below threshold case.Mem. Amer. Math. Soc., 266(1294):v+158, 2020

    work page 2020

  8. [5]

    Bedrossian, P

    J. Bedrossian, P. Germain, and N. Masmoudi. Dynamics near the subcritical transition of the 3D Couette flow II: Above threshold case.Mem. Amer. Math. Soc., 279(1377):v+135, 2022

    work page 2022

Show all 37 references
  1. [6]

    Bedrossian, S

    J. Bedrossian, S. He, S. Iyer, and F. Wang. Stability threshold of nearly-couette shear flows with navier boundary conditions in 2d.arXiv preprint arXiv:2311.00141, 2023

    2023 arXiv

  2. [7]

    Bedrossian, S

    J. Bedrossian, S. He, S. Iyer, and F. Wang. Pseudo-gevrey smoothing for the passive scalar equations near couette.arXiv preprint arXiv:2405.19233, 2024

    2024 arXiv

  3. [8]

    Bedrossian, S

    J. Bedrossian, S. He, S. Iyer, and F. Wang. Uniform inviscid damping and inviscid limit of the 2d navier-stokes equation with navier boundary conditions.arXiv preprint arXiv:2405.19249, 2024

    2024 arXiv

  4. [9]

    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(3):1087–1159, 2016. 29

    2016

  5. [10]

    Bedrossian, V

    J. Bedrossian, V. Vicol, and F. Wang. The Sobolev stability threshold for 2D shear flows near Couette.J. Nonlinear Sci., 28(6):2051–2075, 2018

    2018

  6. [11]

    Q. Chen, S. Ding, Z. Lin, and Z. Zhang. Nonlinear stability for 3-d plane poiseuille flow in a finite channel.arXiv preprint arXiv:2310.11694, 2023

    2023 arXiv

  7. [12]

    Q. Chen, T. Li, D. Wei, and Z. Zhang. Transition threshold for the 2-d couette flow in a finite channel.Archive for Rational Mechanics and Analysis, 238(1):125–183, May 2020

    2020

  8. [13]

    Q. Chen, D. Wei, and Z. Zhang. Transition threshold for the 3d couette flow in a finite channel. Memoirs of the American Mathematical Society, 296, 04 2024

    2024

  9. [14]

    Chen and R

    T. Chen and R. Zi. On the sobolev stability threshold for shear flows near couette in 2d mhd equations.Proceedings of the Royal Society of Edinburgh: Section A Mathematics, pages 1–51, 02 2024

    2024

  10. [15]

    Coti Zelati, T

    M. Coti Zelati, T. M. Elgindi, and K. Widmayer. Enhanced dissipation in the navier–stokes equations near the poiseuille flow.Communications in Mathematical Physics, 378(2):987–1010, July 2020

    2020

  11. [16]

    Del Zotto

    A. Del Zotto. Enhanced dissipation and transition threshold for the poiseuille flow in a periodic strip.SIAM Journal on Mathematical Analysis, 55(5):4410–4424, 2023

    2023

  12. [17]

    Ding and Z

    S. Ding and Z. Lin. Enhanced dissipation and transition threshold for the 2-d plane poiseuille flow via resolvent estimate.arXiv preprint arXiv:2008.10057, 2020

    2008 arXiv

  13. [18]

    M. Dolce. Stability threshold of the 2d couette flow in a homogeneous magnetic field using symmetric variables.Communications in Mathematical Physics, 405(4):94, 2024

    2024

  14. [19]

    Dolce, N

    M. Dolce, N. Knobel, and C. Zillinger. Large norm inflation of the current in the viscous, non-resistive magnetohydrodynamics equations.arXiv preprint arXiv:2410.22804, 2024

    2024 arXiv

  15. [20]

    Gallet, M

    B. Gallet, M. Berhanu, and N. Mordant. Influence of an external magnetic field on forced turbulence in a swirling flow of liquid metal.Physics of Fluids, 21(8), 2009

    2009

  16. [21]

    J. Jin, X. Ren, and D. Wei. Stability threshold of couette flow for 2d mhd equations in the sobolev space.preprint, 2024

    2024

  17. [22]

    N. Knobel. Sobolev stability for the 2d mhd equations in the non-resistive limit.arXiv preprint arXiv:2401.12548, 2024

    2024 arXiv

  18. [23]

    Knobel and C

    N. Knobel and C. Zillinger. On the sobolev stability threshold for the 2d mhd equations with horizontal magnetic dissipation.arXiv preprint arXiv:2309.00496, 2023

    2023 arXiv

  19. [24]

    H. Li, N. Masmoudi, and W. Zhao. Asymptotic stability of two-dimensional couette flow in a viscous fluid.arXiv preprint arXiv:2208.14898, 2022

    2022 arXiv

  20. [25]

    Li and W

    H. Li and W. Zhao. Asymptotic stability in the critical space of 2d monotone shear flow in the viscous fluid.arXiv preprint arXiv:2306.03555, 06 2023. 30

    2023 arXiv

  21. [26]

    T. Li, D. Wei, and Z. Zhang. Pseudospectral bound and transition threshold for the 3d kolmogorov flow.Communications on Pure and Applied Mathematics, 73(3):465–557, 2020

    2020

  22. [28]

    Masmoudi and W

    N. Masmoudi and W. Zhao. Stability threshold of the 2d couette flow in sobolev spaces.arXiv preprint arXiv:1908.11042, 2019

    1908 arXiv

  23. [29]

    S. A. Orszag and L. C. Kells. Transition to turbulence in plane poiseuille and plane couette flow.Journal of Fluid Mechanics, 96(1):159–205, 1980

    1980

  24. [32]

    Wei and Z

    D. Wei and Z. Zhang. Transition threshold for the 3D Couette flow in Sobolev space.Comm. Pure Appl. Math., 74(11):2398–2479, 2021

    2021

  25. [33]

    Wei and Z

    D. Wei and Z. Zhang. Nonlinear enhanced dissipation and inviscid damping for the 2d couette flow.Tunisian Journal of Mathematics, 5(3):573–592, 2023

    2023

  26. [34]

    Wei and Z

    D. Wei and Z. Zhang. Optimal stability threshold of the 2D Couette flow in a finite channel. preprint, 2023

    2023

  27. [35]

    D. Wei, Z. Zhang, and W. Zhao. Linear inviscid damping and enhanced dissipation for the kolmogorov flow.Advances in Mathematics, 362:106963, 2020

    2020

  28. [36]

    C. Zhai, Z. Zhang, and W. Zhao. Long-time behavior of alfv´ en waves in a flowing plasma: generation of the magnetic island.Archive for Rational Mechanics and Analysis, 242:1317– 1394, 2021

    2021

  29. [37]

    Zhao and R

    W. Zhao and R. Zi. Asymptotic stability of couette flow in a strong uniform magnetic field for the euler-mhd system.Archive for Rational Mechanics and Analysis, 248, 05 2024. 31

    2024

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