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Nonlinear Evolution Toward the Linear Diffusive Profile in the Presence of Couette Flow

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Pith's one-line read Large vorticity perturbations near planar Couette flow return to the linear diffusive profile at the optimal $t^{-1/2}$ rate, after a long but explicit time depending on the relative Reynolds number.

desk verdict Real quantitative convergence theorem near Couette, but the abstract promises point-vortex and L1 coverage that the theorem hypotheses do not deliver. read the letter →

arxiv 2505.08486 v1 pith:YQ5MGKUT submitted 2025-05-13 math.AP

Ning Liu , Ping Zhang , Weiren Zhao This is my paper · ORCID
classification math.AP MSC 35Q3035B4035B3576D05
keywords CouetteflowNavier-Stokesequationsvorticitylong-timeasymptoticslargedataFokker-Plancksemigroupenhanceddissipationinvisciddamping
topics Navier-Stokes Regularity
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 claims that the vorticity perturbation of the two-dimensional Navier-Stokes equations near planar Couette flow, starting from large and only mildly regular data, converges to a universal long-time profile: a constant multiple of the fundamental solution $G_L(t)$ of the linearized vorticity equation, with the optimal decay rate $t^{-1/2}$ in $L^1$. The convergence is quantitative and occurs after an explicit time $T_1$ that depends only on the relative Reynolds number $1+\nu^{-1}\|\Omega(1)\|_{L^2(m_0)}$; this is what allows the initial perturbation to be large. The proof is built on a new set of long-time coordinates adapted to the exact Couette kernel, in which the nonlinearity acquires a $\langle t\rangle^{-2}$ smallness even though the data are not small. The advertised singular cases such as point vortices are not covered by the stated theorems, which require the rescaled data at $t=1$ to lie in $L^2(m_0)$ with $m_0>7m/2$.

What carries the argument

The load-bearing construction is a time-dependent change of variables (1.8)-(1.9) built from the exact fundamental solution $G_L(t)$ of the linearized equation (1.3). It rescales the sheared coordinates by the spreading rates of $G_L$: $X=x/\sqrt{\nu t(1+t^2/3)}$ and $Y=((1+t^2/3)y-(t/2)x)/\sqrt{\nu t(1+t^2/3)(1+t^2/12)}$, and writes $\omega(t,x,y)=\Omega(t,X,Y)/(\nu t\sqrt{1+t^2/12})$. In these coordinates the equation becomes $t\partial_t\Omega=L_t\Omega+N_t\Omega$, where $L_t$ converges to the Fokker-Planck operator $L_\infty=4\partial_Y^2+2Y\partial_Y+2+\frac{\sqrt{3}}{2}(X\partial_Y-Y\partial_X)$ whose Gaussian steady state is $G$, and the nonlinearity $N_t$ carries a prefactor $\nu^{-1}(1+t^2/12)^{-1}$, hence decays like $t^{-2}$. The proof combines an explicit Fourier characteristic formula for $e^{\tau L_\infty}$ (which gives exponential decay on weighted $L^2(m)$) with a hypoelliptic energy functional whose successively ordered logarithmic weights propagate $\partial_X$ and $\partial_Y$ derivatives.

What would settle it

Fix any positive viscosity (for instance $\nu=1$) and choose a datum $\Omega(1)\in L^2(m_0)$ satisfying the theorem with a very large relative Reynolds number $1+\nu^{-1}\|\Omega(1)\|_{L^2(m_0)}$. If an accurate numerical integration of the perturbed vorticity equation shows that $\sup_{t\ge T_1} t^{1/2}\|\Omega(t)-M(\Omega(1))G\|_{L^2(m)}$ grows without bound as the datum size increases, or that a two-vortex initial profile leaves a permanent two-hump residue rather than collapsing to a single Gaussian, then (1.17) and Remark 1.4 would be false.

Watch

Extended reading notes

Core claim

On the paper's own terms, the main discovery is Theorem 1.3: for $3\le m$, $m_0>\frac{7}{2}m$, and $\Omega(1)\in L^2(m_0)$, the rescaled vorticity satisfies $\|\Omega(t)-M(\Omega(1))G\|_{L^2(m)}\le C t^{-1/2}(1+\nu^{-1}\|\Omega(1)\|_{L^2(m_0)})^{\frac{8(m+1)}{1-7m/2m_0}-1+\epsilon}\|\Omega(1)\|_{L^2(m_0)}$ for all $t\ge T_1$, where $T_1$ has the same structure with exponent $\frac{7(m+1)}{1-7m/2m_0}+\epsilon$. In the original variables this gives $\|\omega(t)-M(\omega(1))G_L(t)\|_{L^1}=O(t^{-1/2})$ (Remark 1.4), so the nonlinear solution returns to the self-similar diffusive profile of the linearized problem, carrying the same total vorticity mass. The argument is not perturbative: the theorem holds for large data, and the price paid is the long explicit waiting time $T_1$ and the polynomial weight requirement $L^2(m_0)$.

Load-bearing premise

The load-bearing premise is that the rescaled vorticity at time $t=1$ is square-integrable with enough polynomial spatial decay, specifically $\Omega(1)\in L^2(m_0)$ with $m_0>7m/2$ for some $m\ge 3$; without that, the quantitative $O(t^{-1/2})$ convergence in the paper is not proved.

Editorial extensions

If this is right

  • The nonlinear vorticity near Couette flow therefore relaxes, in $L^1$, to $M(\omega(1))G_L(t)$ at the same $t^{-1/2}$ rate as the linearized flow, so the background shear does not change the universal late-time Gaussian character.
  • The waiting time before the $t^{-1/2}$ regime is explicit and finite, so the result is a large-data asymptotic statement rather than a limiting one: convergence is guaranteed after a computable time $T_1$ depending on the relative Reynolds number.
  • Because the limit profile $G_L$ itself exhibits enhanced dissipation and inviscid damping, the theorem implies these mechanisms prevail in the final stage even for large data.
  • Imposing the extra smallness $\nu^{-1}\|\Omega(1)\|_{L^2(m_0)}\lesssim 1$ reproduces the transition-threshold condition $\|\omega(1)\|_{L^2}\lesssim \nu^{1/2}$, matching the known $\beta=1/2$ threshold in nearly $L^2$ norms.
  • Theorem 1.1 supplies the companion existence theory: every $L^1$ vorticity datum has a unique global mild solution, with $L^p$ decay $\|\omega(t)\|_{L^p}\lesssim(\nu t)^{1/p-1}\|\omega_0\|_{L^1}$.

Reading between the lines

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

  • The self-similar variables are built from the exact linear kernel $G_L$, so the same strategy may extend to measure-valued initial vorticity (such as a point vortex) once one shows such a solution enters $L^2(m_0)$ after a finite positive time; the paper itself leaves this as an open problem.
  • The exponents in $T_1$ are far from sharp; for $m=3$ and large $m_0$ they give a timescale of order $(1+\nu^{-1}\|\Omega(1)\|)^{62}$, which suggests numerical experiments on high-Reynolds Couette flow could test whether convergence is actually much faster.
  • One could try to replace the polynomial weight $L^2(m_0)$ by a tail smallness condition on $\Omega(1)$; if that succeeds, the theorem would cover genuinely $L^1$ data that only develop weighted integrability after a short time.
  • The explicit Fourier representation of $e^{\tau L_\infty}$ is a transferable tool: analogous characteristics could give quantitative convergence to self-similar profiles for other shear flows with explicit linear kernels.
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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

2 major / 5 minor

Summary. The paper studies the 2D Navier-Stokes equations near planar Couette flow in the whole plane. It first establishes global well-posedness of the vorticity formulation for arbitrary L^1 initial data (Theorem 1.1), with heat-like L^p decay. For the rescaled vorticity in variables adapted to the explicit fundamental solution G_L, it proves weighted L^2(m) growth estimates and higher-derivative bounds (Theorem 1.2), and then shows that, for initial data at time t=1 in a sufficiently strong weighted space, the vorticity converges to the mass times the Gaussian fundamental solution of the linearized problem at rate t^{-1/2}, after an explicit but very large time T_1 (Theorem 1.3). The proof combines a change of variables based on G_L, hypocoercive energy estimates with logarithmic coefficients, and an explicit Fourier solution formula for the limiting Fokker-Planck operator L_∞. The abstract advertises coverage of general perturbations 'including singular configurations such as point vortices'; this advertised scope is not supported by the stated theorems.

Significance. If the main theorem is correct as stated, it constitutes a quantitative, large-data asymptotic stability result for Couette flow in R^2, with no smallness assumption on the perturbation and with an explicit (though extremely large) waiting time. Notable strengths include the explicit Fourier formula for the semigroup generated by L_∞, the clean derivation of the decay rate e^{-\tau/2} in weighted spaces, and the transparent parameter dependence in the bounds. The paper also gives a useful comparison with the transition-threshold literature and with the prior work of Gallay-Wayne. However, the actual theorems require the rescaled vorticity at t=1 to belong to L^2(m_0) with m_0 > 7m/2; this is a genuine spectral/decay condition that is not implied by the L^1 well-posedness theory. The advertised inclusion of point vortices is explicitly left open in Sections 1.2 and Remark 1.6, so the abstract overstates the scope of the results.

major comments (2)
  1. [Abstract and §1.2, Remark 1.6] The abstract claims that the asymptotic convergence holds for 'general perturbations, which may be large and of low regularity, including singular configurations such as point vortices.' This claim is not a consequence of any theorem in the paper. Theorem 1.3 requires \Omega(1)\in L^2(m_0) with m_0 > 7m/2 (m\ge 3), and the point-vortex case is explicitly identified as an open problem in §1.2 and Remark 1.6. Moreover, the L^1 well-posedness of Theorem 1.1 does not imply the weighted hypothesis: for example, f(z)=(1+|z|^2)^{-3/2}\in L^1(R^2) evolves to a profile with a |z|^{-3} tail at t=1 under the linear semigroup, so the rescaled function lies in L^2(m_0) only for m_0<2, not for m_0\ge 3. The abstract and introduction should be revised to state the actual hypothesis \Omega(1)\in L^2(m_0), and the point-vortex claim should be removed or clearly qualified as an open problem.
  2. [Section 3, beginning and Proposition 3.1] The proofs of Theorems 1.2 and 1.3 are based on a priori estimates for smooth solutions of (1.10), as stated at the start of Section 3, but no approximation or regularization argument is provided to justify that the mild solution constructed in Theorem 1.1 (with only L^1 initial data) satisfies these estimates. Since the paper presents these as unconditional theorems for the unique solution, the authors should either prove that the solution is smooth for positive times (which is plausible by parabolic smoothing and the Biot-Savart law) or add a standard density/approximation argument with uniform constants. As written, this is a gap in the proof of the central convergence theorem.
minor comments (5)
  1. [Section 2.2, before Lemma 2.2] The text says the L^1 norm is 'non-decreasing' with respect to time, but Lemma 2.2 and the surrounding discussion prove that it is non-increasing; the word should be corrected.
  2. [Section 1.2, first paragraph] There is a typo: 'Coutte flow' should be 'Couette flow'.
  3. [Theorem 1.1, estimate (1.7)] The constant C in (1.7) appears to depend on p, since the proof via dyadic iteration produces p-dependent constants; the statement should explicitly say C=C(p) to avoid confusion.
  4. [Section 4.2, after (4.25)] The phrase 'the other powers are all negative' is somewhat misleading: the exponents in (4.20) and (4.21) contain positive powers of t_0 in the displayed constants, and their decay in t_0 comes from the chosen small parameters; this is presumably correct but the phrasing should be clarified.
  5. [Remark 1.5] The numerical value 'nearly T_2 \approx (1+\nu^{-1}\|\Omega(1)\|)^{62}' is a useful illustration, but the phrase 'nearly' is informal; it would be clearer to state that the exponent is obtained in the limit m=3, m_0\to\infty, and that all terms depending on \delta,\sigma,\epsilon are absorbed into the stated 'nearly'.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the asymptotic profile is derived from the linearized semigroup and nonlinear terms are controlled by a priori estimates, not assumed.

full rationale

The paper's central claim, Theorem 1.3, is derived by a direct perturbative argument: the solution is written in the rescaled variables (1.8)-(1.9), the target Gaussian profile G is the kernel of the limit operator L_infty defined in (1.12), and the semigroup decay (1.14) is proved in Proposition 4.1 from an explicit Fourier solution formula (4.3)-(4.5), without invoking the target result as an input. The nonlinear terms are then bounded through the a priori weighted estimates (1.15)-(1.16) and the small factor <t>^{-2} in (1.11), leading to the quantitative convergence (1.17). No parameter is fitted to the data, and no conclusion is inserted as an assumption beyond the stated weighted-L2 regularity of Omega(1). The rescaling (1.8) is motivated by the explicit fundamental solution G_L, but the final statement in original variables, Remark 1.4, is a genuine convergence result rather than a tautology. The paper also candidly states its own limitation in Section 1.2 and Remark 1.6: it assumes Omega(1) in L^2(m0) and leaves the long-time behavior of general L^1 solutions, including point-vortex data, as an open problem. This scope gap concerns the abstract's advertised coverage, not circularity of the derivation. The only self-citation, reference [33], is used for comparison with the transition threshold and is not load-bearing for the main theorem. The derivation is self-contained against the explicit linear semigroup and standard energy estimates, so no significant circularity is present.

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

No free parameters are fitted to data; the theorem is parameter-free in its qualitative content, and the constants m, m0, delta, sigma, epsilon are technical parameters that can be chosen within stated ranges. The only assumptions are standard PDE tools and the technical L^2(m0) condition at time t=1, which is a genuine restriction relative to the abstract.

assumptions (4)
  • standard math Classical Fourier analysis, Sobolev embedding, and the 2D Biot-Savart law including ||nabla Delta^{-1} omega||_{L^infty} bounds.
    Used throughout Section 3, e.g. Lemma 3.1 and Lemma 3.3, to control velocity terms in the rescaled variables.
  • standard math Hypoellipticity and spectral theory of the limiting Fokker-Planck operator L_infty on Gaussian weighted L^2(G).
    Invoked for (1.13) and proved for L^2(m) in Proposition 4.1 using known results from Helffer-Nier [25] and Helffer [24].
  • standard math Global well-posedness theory for the 2D Navier-Stokes vorticity equation with L^1 initial data, following Kato, Ben-Artzi, and Brezis.
    Used for Theorem 1.1 and the mild solution formulation (2.1), citing [8, 10, 30].
  • domain assumption The rescaled initial data at time t=1 belongs to L^2(m0) with m0 > 7m/2, rather than at t=0.
    The change of variables (1.8) is singular at t=0; Theorem 1.3 assumes Omega(1) in L^2(m0), as stated in Remark 1.6. This is a genuine restriction relative to the abstract's L^1 and point-vortex language.

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Pith. "Pith review of Nonlinear Evolution Toward the Linear Diffusive Profile in the Presence of Couette Flow." pith.science (2026). https://pith.science/paper/YQ5MGKUT

@misc{pith2026250508486,
  author       = {Pith},
  title        = {Pith review of: Nonlinear Evolution Toward the Linear Diffusive Profile in the Presence of Couette Flow},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YQ5MGKUT}},
  note         = {Machine review of arXiv:2505.08486}
}
read the original abstract

In this paper, we investigate the long-time behavior of solutions to the two-dimensional Navier-Stokes equations with initial data evolving under the influence of the planar Couette flow. We focus on general perturbations, which may be large and of low regularity, including singular configurations such as point vortices, and show that the vorticity asymptotically approaches a constant multiple of the fundamental solution of the corresponding linearized vorticity equation after a long-time evolution determined by the relative Reynolds number.

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Works this paper leans on

46 extracted references · 42 canonical work pages

  1. [1]

    Albritton, R

    D. Albritton, R. Beekie and M. Novack, Enhanced dissipation and H¨ ormander’s hypoellipticity,J. Funct. Anal.,283(3) (2022), 109522

    work page 2022

  2. [2]

    Arbon and J

    R. Arbon and J. Bedrossian, Quantitative hydrodynamic stability for Couette flow on unbounded domains with Navier boundary conditions,arXiv:2404.02412(2024)

    arXiv 2024

  3. [3]

    Bahouri, J.-Y

    H. Bahouri, J.-Y. Chemin and R. Danchin,Fourier Analysis and Nonlinear Partial Differential Equa- tions, Grundlehren der mathematischen Wissenschaften,343, Springer-Verlag Berlin Heidelberg, 2011

    work page 2011

  4. [4]

    Bedrossian, P

    J. Bedrossian, P. Germain and N. Masmoudi, On the stability threshold for the 3D Couette flow in Sobolev regularity,Ann. Math.,185(2) (2017), 541–608

    work page 2017

  5. [5]

    Bedrossian and N

    J. Bedrossian and N. Masmoudi, Inviscid damping and the asymptotic stability of planar shear flows in the 2D Euler equations,Publ. Math. Inst. Hautes ´Etudes Sci.,122(1) (2015), 195–300

    work page 2015

  6. [6]

    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) (2016), 1087–1159

    work page 2016

  7. [7]

    Bedrossian, V

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

    work page 2018

  8. [8]

    Ben-Artzi, Global solutions of two-dimensional Navier-Stokes and Euler equations,Arch

    M. Ben-Artzi, Global solutions of two-dimensional Navier-Stokes and Euler equations,Arch. Ration. Mech. Anal.,128(4) (1994), 329–358

    work page 1994

Show all 46 references
  1. [9]

    Bottin, O

    S. Bottin, O. Dauchot, F. Daviaud and P. Manneville, Experimental evidence of streamwise vortices as finite amplitude solutions in transitional plane Couette flow,Phys. Fluids,10(10) (1998), 2597–2607

    1998

  2. [10]

    Global solutions of two-dimensional Navier-Stokes and Euler equations

    H. Brezis, Remarks on the preceding paper by M. Ben-Artzi “Global solutions of two-dimensional Navier-Stokes and Euler equations”,Arch. Ration. Mech. Anal.,128(4) (1994), 359–360

    1994

  3. [11]

    Carpio, Asymptotic behavior for the vorticity equations in dimensions two and three,Comm

    A. Carpio, Asymptotic behavior for the vorticity equations in dimensions two and three,Comm. Partial Differ. Equ.,19(5-6) (1994), 827–872

    1994

  4. [12]

    Q. Chen, D. Wei, P. Zhang, and Z. Zhang, Nonlinear inviscid damping for 2-D inhomogeneous incom- pressible Euler equations.J. Eur. Math. Soc.(2025). NONLINEAR EVOLUTION TOWARD THE LINEAR DIFFUSIVE PROFILE 43

    2025

  5. [13]

    Cottet, ´Equations de Navier-Stokes dans le plan avec tourbillon initial mesure,C

    G-H. Cottet, ´Equations de Navier-Stokes dans le plan avec tourbillon initial mesure,C. R. Acad. Sci. Paris S´ er. I Math.,303(4) (1986), 105–108

    1986

  6. [14]

    Couliou and R

    M. Couliou and R. Monchaux, Growth dynamics of turbulent spots in plane Couette flow,J. Fluid Mech.,819(2017), 1–20

    2017

  7. [15]

    Dauchot and F

    O. Dauchot and F. Daviaud, Streamwise vortices in plane Couette flow,Phys. Fluids,7(5) (1995), 901–903

    1995

  8. [16]

    Deng and N

    Y. Deng and N. Masmoudi, Long-time instability of the Couette Flow in Low Gevrey Spaces,Comm. Pure Appl. Math.,76(10) (2023), 2804–2887

    2023

  9. [17]

    Donati and T

    M. Donati and T. Gallay, Fast relaxation of a viscous vortex in an external flow,arXiv:2503.20643 (2025)

    2025

  10. [18]

    Gallagher and T

    I. Gallagher and T. Gallay, Uniqueness for the two-dimensional Navier–Stokes equation with a measure as initial vorticity,Math. Ann.,332(2) (2005), 287–327

    2005

  11. [19]

    Gallagher, T

    I. Gallagher, T. Gallay and P.-L. Lions, On the uniqueness of the solution of the two-dimensional Navier–Stokes equation with a Dirac mass as initial vorticity,Math. Nachr.,278(14) (2005), 1665– 1672

    2005

  12. [20]

    Gallay and C

    T. Gallay and C. E. Wayne, Invariant manifolds and the long-time asymptotics of the Navier-Stokes and vorticity equations onR 2,Arch. Ration. Mech. Anal.,163(3) (2002), 209–258

    2002

  13. [21]

    Gallay and C

    T. Gallay and C. E. Wayne, Global stability of vortex solutions of the two-dimensional Navier-Stokes equation,Comm. Math. Phys.,255(1) (2005), 97–129

    2005

  14. [22]

    Giga and T

    Y. Giga and T. Kambe, Large time behavior of the vorticity of two-dimensional viscous flow and its application to vortex formation,Comm. Math. Phys.,117(4) (1988), 549–568

    1988

  15. [23]

    Y. Giga, T. Miyakawa and H. Osada, Two-dimensional Navier-Stokes flow with measures as initial vorticity,Arch. Ration. Mech. Anal.,104(3) (1988), 223–250

    1988

  16. [24]

    Helffer,Spectral theory and its applications, Cambridge Studies in Advanced Mathematics,139, Cambridge University Press, Cambridge, 2013

    B. Helffer,Spectral theory and its applications, Cambridge Studies in Advanced Mathematics,139, Cambridge University Press, Cambridge, 2013

    2013

  17. [25]

    Helffer and F

    B. Helffer and F. Nier,Hypoelliptic estimates and spectral theory for Fokker-Planck operators and Witten Laplacians, Lecture Notes in Mathematics,1862, Springer-Verlag, Berlin, 2005

    2005

  18. [26]

    H¨ ormander, Hypoelliptic second order differential equations,Acta Math.,119(1) (1967), 147–171

    L. H¨ ormander, Hypoelliptic second order differential equations,Acta Math.,119(1) (1967), 147–171

    1967

  19. [27]

    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. [28]

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

    2023

  21. [29]

    Kato, StrongL p-solutions of the Navier-Stokes equation inRm, with applications to weak solutions, Math

    T. Kato, StrongL p-solutions of the Navier-Stokes equation inRm, with applications to weak solutions, Math. Z.,187(1984), 471–480

    1984

  22. [30]

    Kato, The Navier-Stokes equation for an incompressible fluid inR 2 with a measure as the initial vorticity,Differ

    T. Kato, The Navier-Stokes equation for an incompressible fluid inR 2 with a measure as the initial vorticity,Differ. Integral Equ.,7(4) (1994), 949–966

    1994

  23. [31]

    Kelvin, Stability of fluid motion: rectilinear motion of viscous fluid between two parallel plates, Philos

    L. Kelvin, Stability of fluid motion: rectilinear motion of viscous fluid between two parallel plates, Philos. Mag.,24(5) (1887), 188–196

  24. [32]

    Leray, ´Etude de diverses ´ equations int´ egrales non lin´ eaires et de quelques probl` emes que pose l’hydrodynamique,J

    J. Leray, ´Etude de diverses ´ equations int´ egrales non lin´ eaires et de quelques probl` emes que pose l’hydrodynamique,J. Math. Pures Appl.,12(1933), 1–82

    1933

  25. [33]

    H. Li, N. Liu and W. Zhao, Stability threshold of the two-dimensional Couette flow in the whole plane, arXiv:2501.10818(2025)

    2025 arXiv

  26. [34]

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

    2022 arXiv

  27. [35]

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

    2024

  28. [36]

    Masmoudi and W

    N. Masmoudi and W. Zhao, Enhanced dissipation for the 2D Couette flow in critical space,Comm. Partial Differ. Equ.,45(12) (2020), 1682–1701

    2020

  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 Anal. Non Lin´ eaire,39(2) (2022), 245–325

    2022

  30. [38]

    Masmoudi and W

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

    2024

  31. [39]

    W. M. Orr, The stability or instability of the steady motions of a perfect liquid and of a viscous liquid, Proc. R. Ir. Acad. Sect. A,27(9) (1907), 9–68

    1907

  32. [40]

    Rayleigh, On the Stability, or Instability, of certain Fluid Motions,Proc

    L. Rayleigh, On the Stability, or Instability, of certain Fluid Motions,Proc. Lond. Math. Soc.,1(1) (1879/80), 57–70. 44 N. LIU, P. ZHANG, AND W. ZHAO

  33. [41]

    Sommerfeld, Ein Beitrag zur hydrodynamischen Erkl¨ arung der turbulenten Fl¨ ussigkeitsbewegung, Atti del IV Congresso internazionale dei matematici(1908), 116–124

    A. Sommerfeld, Ein Beitrag zur hydrodynamischen Erkl¨ arung der turbulenten Fl¨ ussigkeitsbewegung, Atti del IV Congresso internazionale dei matematici(1908), 116–124

    1908

  34. [42]

    Villani, Hypocoercivity,Mem

    C. Villani, Hypocoercivity,Mem. Amer. Math. Soc.,202(950) (2009), iv+141 pp

    2009

  35. [43]

    Wang and W

    G. Wang and W. Wang, Transition threshold for the 2-D Couette flow in whole space via Green’s function,arXiv:2404.11878(2024)

    2024 arXiv

  36. [44]

    Wei and Z

    D. Wei and Z. Zhang, Nonlinear enhanced dissipation and inviscid damping for the 2D Couette flow, Tunis. J. Math.,5(3) (2023), 573–592

    2023

  37. [45]

    V. B. Zametaev and A. R. Gorbushin, Evolution of vortices in 2D boundary layer and in the Couette flow,AIP Conf. Proc.,1770(1) (2016)

    2016

  38. [46]

    Zhao, Inviscid damping of monotone shear flows for 2D inhomogeneous Euler equation with non- constant density in a finite channel ,Ann

    W. Zhao, Inviscid damping of monotone shear flows for 2D inhomogeneous Euler equation with non- constant density in a finite channel ,Ann. PDE11.1(2025): 8. (N. Liu)Academy of Mathematics&Systems Science, The Chinese Academy of Sciences, Bei- jing 100190, CHINA. Email address:...

    2025

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