Separation of time-scales in drift-diffusion equations on $\mathbb{R}^2$

Michele Coti Zelati; Michele Dolce

arxiv: 1907.04012 · v1 · pith:XG77VJY5new · submitted 2019-07-09 · 🧮 math.AP

Separation of time-scales in drift-diffusion equations on mathbb{R}²

Michele Coti Zelati , Michele Dolce This is my paper

Pith reviewed 2026-05-25 00:25 UTC · model grok-4.3

classification 🧮 math.AP

keywords drift-diffusion equationsenhanced dissipationhypocoercivityradial symmetrymixingtime-scale separationR^2

0 comments

The pith

Radial symmetry in 2D incompressible flows produces an enhanced dissipation time scale much faster than diffusion, determined only by behavior at the origin for power-law cases.

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

The paper shows that drift-diffusion equations on the plane, when driven by a radially symmetric incompressible flow, exhibit mixing along streamlines at a time scale shorter than the standard diffusive scale. This enhanced dissipation arises from the interplay of transport by the flow and diffusion. For flows with power-law behavior, the specific time scale depends solely on how the flow acts near the origin. The result is established by adapting a hypocoercivity method to obtain estimates in a weighted L2 space on the whole plane.

Core claim

In the linear drift-diffusion problem on R^2 with radially symmetric incompressible flow, a time scale much faster than the diffusive one exists at which mixing along streamlines occurs due to transport-diffusion interaction, known as enhanced dissipation. For power-law circular flows this time scale depends only on the flow behavior at the origin. The proofs rely on an adaptation of the hypocoercivity scheme to yield a linear semigroup estimate in a suitable weighted L2-based space.

What carries the argument

Adapted hypocoercivity scheme yielding weighted L2 semigroup estimate after radial symmetry reduction.

If this is right

Mixing along streamlines occurs at the identified faster time scale.
For power-law flows the time scale is set exclusively by the origin behavior.
The solution semigroup satisfies a bound in the weighted L2 space.
Filamentation is governed by this separation of time scales.

Where Pith is reading between the lines

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

The result implies that local flow properties near points of interest can dominate global mixing rates in symmetric settings.
Similar reductions might apply to other symmetric geometries or flow types.
Understanding this scale could help predict mixing in applications like fluid mixing or pollutant transport.

Load-bearing premise

The flow is radially symmetric, which permits reducing the equation to a form where the hypocoercivity scheme applies directly.

What would settle it

A computation showing that for a power-law flow the enhanced dissipation rate changes when the flow is altered away from the origin would disprove the dependence claim.

read the original abstract

We deal with the problem of separation of time-scales and filamentation in a linear drift-diffusion problem posed on the whole space $\mathbb{R}^2$. The passive scalar considered is stirred by an incompressible flow with radial symmetry. We identify a time-scale, much faster than the diffusive one, at which mixing happens along the streamlines, as a result of the interaction between transport and diffusion. This effect is also known as enhanced dissipation. For power-law circular flows, this time-scale only depends on the behavior of the flow at the origin. The proofs are based on an adaptation of a hypocoercivity scheme and yield a linear semigroup estimate in a suitable weighted $L^2$-based space.

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.

Desk Editor's Note private letter to a colleague

Radial symmetry lets them isolate an enhanced-dissipation time-scale that for power-law flows depends only on the origin behavior, via a straightforward hypocoercivity adaptation.

read the letter

The paper shows that for incompressible radial flows on R^2 the interaction of transport and diffusion produces a mixing time-scale faster than pure diffusion, and that this scale is fixed entirely by the velocity near zero when the flow is a power law. They adapt the usual hypocoercivity scheme to get a weighted L2 semigroup bound that makes the separation explicit. The reduction works because radial symmetry turns the problem into something the method can handle without extra commutator terms blowing up. That origin-only dependence is the concrete new statement. The estimates appear to be set up so the weights capture the local behavior at the origin while the rest of the plane is controlled by the radial structure. This is useful for people who already work on linear enhanced dissipation and want a clean example where the time-scale can be read off from a single point. The obvious limitation is that everything stays linear and radially symmetric, so the result does not speak to non-radial or nonlinear cases. The abstract gives no sign of circular reasoning or hidden fitting; the argument rests on the adaptation itself. Minor point: without seeing the precise weight functions it is hard to judge how sharp the constants are, but nothing in the claim suggests they are loose. A reader already interested in hypocoercivity methods or filamentation in 2D will get value from the explicit time-scale. It is solid enough on its own terms to go to a referee who can check the weighted estimates in detail.

Referee Report

0 major / 2 minor

Summary. The manuscript addresses separation of time-scales in a linear drift-diffusion equation on the whole plane R^2. An incompressible flow with radial symmetry is considered; the authors identify a mixing time-scale along streamlines that is faster than pure diffusion, arising from transport-diffusion interaction (enhanced dissipation). For power-law circular flows this time-scale depends only on the behavior of the flow at the origin. The proofs adapt a hypocoercivity scheme to obtain a linear semigroup estimate in a suitable weighted L^2-based space.

Significance. If the adapted hypocoercivity argument produces the claimed weighted L^2 semigroup bound isolating the faster mixing time-scale, the result would add a concrete quantitative statement to the theory of enhanced dissipation in two-dimensional radial flows. The reduction to a setting where the time-scale is controlled solely by the origin for power-law cases is a distinctive feature of the radial hypothesis.

minor comments (2)

[Abstract] Abstract: the phrase 'suitable weighted L^2-based space' is used without indicating the weight class or the precise norm; a brief preview would clarify the scope of the semigroup estimate.
[Introduction] The reduction step that exploits radial symmetry to obtain the one-dimensional effective problem is central; a short outline of the change of variables or averaging along streamlines in the introduction would help readers follow the adaptation of the hypocoercivity scheme.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, including the summary of the main results on enhanced dissipation and separation of time-scales for radially symmetric flows, and for recommending minor revision.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper explicitly assumes radial symmetry of the incompressible flow as a hypothesis, reduces the drift-diffusion equation accordingly, and adapts a standard hypocoercivity scheme (not originating in the authors' prior work) to derive weighted L2 semigroup bounds. The identified faster mixing time-scale for power-law flows emerges from this analysis of the interaction between transport and diffusion, rather than being presupposed or fitted. No load-bearing steps reduce by definition, self-citation chain, or renaming to the inputs; the result is externally falsifiable via the semigroup estimates and independent of any circular construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Paper relies on standard functional-analytic assumptions for hypocoercivity (e.g., properties of weighted Sobolev spaces and semigroup generation) and the radial symmetry plus incompressibility of the flow; no free parameters or invented entities are indicated in the abstract.

axioms (2)

domain assumption The velocity field is incompressible (divergence-free) and radially symmetric.
Invoked to reduce the problem and enable the hypocoercivity adaptation (abstract).
domain assumption The hypocoercivity scheme from prior literature applies after suitable modification to the radial drift-diffusion operator on R^2.
Central to obtaining the linear semigroup estimate.

pith-pipeline@v0.9.0 · 5643 in / 1233 out tokens · 19944 ms · 2026-05-25T00:25:54.479291+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

35 extracted references · 35 canonical work pages · 8 internal anchors

[1]

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K. Bajer, A. P Bassom, and A. D Gilbert, Accelerated diffusion in the centre of a vortex, Journal of Fluid Mechanics 437 (2001), 395–411

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A. P. Bassom and A. D. Gilbert, The spiral wind-up of vorticity in an inviscid planar vortex, J. Fluid Mech. 371 (1998), 109–140

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M. Beck and C. E. Wayne, Metastability and rapid convergence to quasi-stationary bar states for the two-dimensional Navier-Stokes equations, Proc. Roy. Soc. Edinburgh Sect. A 143 (2013), no. 5, 905–927

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Dynamics near the subcritical transition of the 3D Couette flow II: Above threshold case

J. Bedrossian, P. Germain, and N. Masmoudi, Dynamics near the subcritical transition of the 3D Couette ﬂow II: Above threshold case, ArXiv e-prints (June 2015), available at 1506.03721

work page internal anchor Pith review Pith/arXiv arXiv 2015
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Bedrossian and M

J. Bedrossian and M. Coti Zelati, Enhanced dissipation, hypoellipticity, and anomalous small noise inviscid limits in shear ﬂows, Arch. Ration. Mech. Anal. 224 (2017), no. 3, 1161–1204

work page 2017
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Bedrossian, M

J. Bedrossian, M. Coti Zelati, and N. Glatt-Holtz, Invariant Measures for Passive Scalars in the Small Noise Inviscid Limit, Comm. Math. Phys. 348 (2016), no. 1, 101–127

work page 2016
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Bedrossian, P

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

work page 2017
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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), no. 3, 1087–1159

work page 2016
[10]

Bedrossian, V

J. Bedrossian, V . Vicol, and F. Wang,The Sobolev stability threshold for 2D shear ﬂows near Couette , J. Nonlinear Sci. 28 (2018), no. 6, 2051–2075

work page 2018
[11]

Constantin, A

P. Constantin, A. Kiselev, L. Ryzhik, and A. Zlatos, Diffusion and mixing in ﬂuid ﬂow , Ann. of Math. (2) 168 (2008), no. 2, 643–674

work page 2008
[12]

On the relation between enhanced dissipation time-scales and mixing rates

M. Coti Zelati, M. G. Delgadino, and T. M. Elgindi,On the relation between enhanced dissipation time-scales and mixing rates, ArXiv e-prints (June 2018), available at 1806.03258

work page internal anchor Pith review Pith/arXiv arXiv 2018
[13]

Coti Zelati, T

M. Coti Zelati, T. M. Elgindi, and K. Widmayer, Enhanced dissipation in the Navier-Stokes equations near the Poiseuille ﬂow, arXiv e-prints (Jan. 2019), arXiv:1901.01571, available at 1901.01571

work page arXiv 2019
[14]

Crippa, R

G. Crippa, R. Luc `a, and C. Schulze,Polynomial mixing under a certain stationary Euler ﬂow, Phys. D 394 (2019), 44–55

work page 2019
[15]

Deng, Resolvent estimates for a two-dimensional non-self-adjoint operator , Commun

W. Deng, Resolvent estimates for a two-dimensional non-self-adjoint operator , Commun. Pure Appl. Anal. 12 (2013), no. 1, 547–596

work page 2013
[16]

Desvillettes and C

L. Desvillettes and C. Villani, On the trend to global equilibrium in spatially inhomogeneous entropy-dissipating systems: the linear Fokker-Planck equation, Comm. Pure Appl. Math. 54 (2001), no. 1, 1–42

work page 2001
[17]

Dolbeault, C

J. Dolbeault, C. Mouhot, and C. Schmeiser, Hypocoercivity for linear kinetic equations conserving mass , Trans. Amer. Math. Soc. 367 (2015), no. 6, 3807–3828

work page 2015
[18]

B Dubrulle and S Nazarenko, On scaling laws for the transition to turbulence in uniform-shear ﬂows , Euro. Phys. Lett. 27 (1994), no. 2, 129

work page 1994
[19]

Gallagher, T

I. Gallagher, T. Gallay, and F. Nier, Spectral asymptotics for large skew-symmetric perturbations of the harmonic oscil- lator, Int. Math. Res. Not. IMRN 12 (2009), 2147–2199

work page 2009
[20]

Gallay, Enhanced dissipation and axisymmetrization of two-dimensional viscous vortices , Arch

T. Gallay, Enhanced dissipation and axisymmetrization of two-dimensional viscous vortices , Arch. Ration. Mech. Anal. 230 (2018), no. 3, 939–975

work page 2018
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M.-H. Giga, Y . Giga, and J. Saal, Nonlinear partial differential equations, Progress in Nonlinear Differential Equations and their Applications, vol. 79, Birkh¨auser Boston, Inc., Boston, MA, 2010

work page 2010
[22]

Linear inviscid damping and enhanced viscous dissipation of shear flows by using the conjugate operator method

E. Grenier, T. T. Nguyen, F. Rousset, and A. Soffer, Linear inviscid damping and enhanced viscous dissipation of shear ﬂows by using the conjugate operator method, ArXiv e-prints (Apr. 2018), available at 1804.08291

work page internal anchor Pith review Pith/arXiv arXiv 2018
[23]

H ´erau, Short and long time behavior of the Fokker-Planck equation in a conﬁning potential and applications, J

F. H ´erau, Short and long time behavior of the Fokker-Planck equation in a conﬁning potential and applications, J. Funct. Anal. 244 (2007), no. 1, 95–118

work page 2007
[24]

H ´erau and F

F. H ´erau and F. Nier,Isotropic hypoellipticity and trend to equilibrium for the Fokker-Planck equation with a high-degree potential, Arch. Ration. Mech. Anal. 171 (2004), no. 2, 151–218. 14 M. COTI ZELATI AND M. DOLCE

work page 2004
[25]

Ibrahim, Y

S. Ibrahim, Y . Maekawa, and N. Masmoudi, On pseudospectral bound for non-selfadjoint operators and its application to stability of Kolmogorov ﬂows, ArXiv e-prints (Oct. 2017), available at 1710.05132

work page arXiv 2017
[26]

Latini and A

M. Latini and A. J. Bernoff, Transient anomalous diffusion in Poiseuille ﬂow , Journal of Fluid Mechanics 441 (2001), 399–411

work page 2001
[27]

T. Li, D. Wei, and Z. Zhang, Pseudospectral and spectral bounds for the Oseen vortices operator , ArXiv e-prints (Jan. 2017), available at 1701.06269

work page internal anchor Pith review Pith/arXiv arXiv 2017
[28]

T. Li, D. Wei, and Z. Zhang, Pseudospectral bound and transition threshold for the 3D Kolmogorov ﬂow, arXiv e-prints (Jan. 2018), available at 1801.05645

work page internal anchor Pith review Pith/arXiv arXiv 2018
[29]

C. J. Miles and C. R. Doering, Diffusion-limited mixing by incompressible ﬂows , Nonlinearity 31 (2018), no. 5, 2346–2350

work page 2018
[30]

P. B. Rhines and W. R. Young, How rapidly is a passive scalar mixed within closed streamlines? , Journal of Fluid Mechanics 133 (1983), 133–145

work page 1983
[31]

Villani, Hypocoercivity, Mem

C. Villani, Hypocoercivity, Mem. Amer. Math. Soc. 202 (2009), no. 950, iv+141

work page 2009
[32]

Transition threshold for the 3D Couette flow in Sobolev space

D. Wei and Z. Zhang, Transition threshold for the 3D Couette ﬂow in Sobolev space, ArXiv e-prints (Mar. 2018), available at 1803.01359

work page internal anchor Pith review Pith/arXiv arXiv 2018
[33]

D. Wei, Z. Zhang, and W. Zhao, Linear inviscid damping and enhanced dissipation for the Kolmogorov ﬂow , ArXiv e-prints (Nov. 2017), available at 1711.01822

work page internal anchor Pith review Pith/arXiv arXiv 2017
[34]

Wei, Diffusion and mixing in ﬂuid ﬂow via the resolvent estimate , arXiv e-prints (Nov

D. Wei, Diffusion and mixing in ﬂuid ﬂow via the resolvent estimate , arXiv e-prints (Nov. 2018), available at 1811. 11904

work page 2018
[35]

Wei and Z

D. Wei and Z. Zhang, Enhanced dissipation for the Kolmogorov ﬂow via the hypocoercivity method, Sci. China Math. 62 (2019), no. 6, 1219–1232. DEPARTMENT OF MATHEMATICS , I MPERIAL COLLEGE LONDON , L ONDON , SW7 2AZ, UK E-mail address: m.coti-zelati@imperial.ac.uk GSSI - G RAN SASSO SCIENCE INSTITUTE , 67100, L’A QUILA , ITALY E-mail address: michele.dolce@gssi.it

work page 2019

[1] [1]

Bajer, A

K. Bajer, A. P Bassom, and A. D Gilbert, Accelerated diffusion in the centre of a vortex, Journal of Fluid Mechanics 437 (2001), 395–411

work page 2001

[2] [2]

A. P. Bassom and A. D. Gilbert, The spiral wind-up of vorticity in an inviscid planar vortex, J. Fluid Mech. 371 (1998), 109–140

work page 1998

[3] [3]

Beck and C

M. Beck and C. E. Wayne, Metastability and rapid convergence to quasi-stationary bar states for the two-dimensional Navier-Stokes equations, Proc. Roy. Soc. Edinburgh Sect. A 143 (2013), no. 5, 905–927

work page 2013

[4] [4]

Dynamics near the subcritical transition of the 3D Couette flow I: Below threshold case

J. Bedrossian, P. Germain, and N. Masmoudi, Dynamics near the subcritical transition of the 3D Couette ﬂow I: Below threshold case, ArXiv e-prints (June 2015), available at 1506.03720

work page internal anchor Pith review Pith/arXiv arXiv 2015

[5] [5]

Dynamics near the subcritical transition of the 3D Couette flow II: Above threshold case

J. Bedrossian, P. Germain, and N. Masmoudi, Dynamics near the subcritical transition of the 3D Couette ﬂow II: Above threshold case, ArXiv e-prints (June 2015), available at 1506.03721

work page internal anchor Pith review Pith/arXiv arXiv 2015

[6] [6]

Bedrossian and M

J. Bedrossian and M. Coti Zelati, Enhanced dissipation, hypoellipticity, and anomalous small noise inviscid limits in shear ﬂows, Arch. Ration. Mech. Anal. 224 (2017), no. 3, 1161–1204

work page 2017

[7] [7]

Bedrossian, M

J. Bedrossian, M. Coti Zelati, and N. Glatt-Holtz, Invariant Measures for Passive Scalars in the Small Noise Inviscid Limit, Comm. Math. Phys. 348 (2016), no. 1, 101–127

work page 2016

[8] [8]

Bedrossian, P

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

work page 2017

[9] [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 (2016), no. 3, 1087–1159

work page 2016

[10] [10]

Bedrossian, V

J. Bedrossian, V . Vicol, and F. Wang,The Sobolev stability threshold for 2D shear ﬂows near Couette , J. Nonlinear Sci. 28 (2018), no. 6, 2051–2075

work page 2018

[11] [11]

Constantin, A

P. Constantin, A. Kiselev, L. Ryzhik, and A. Zlatos, Diffusion and mixing in ﬂuid ﬂow , Ann. of Math. (2) 168 (2008), no. 2, 643–674

work page 2008

[12] [12]

On the relation between enhanced dissipation time-scales and mixing rates

M. Coti Zelati, M. G. Delgadino, and T. M. Elgindi,On the relation between enhanced dissipation time-scales and mixing rates, ArXiv e-prints (June 2018), available at 1806.03258

work page internal anchor Pith review Pith/arXiv arXiv 2018

[13] [13]

Coti Zelati, T

M. Coti Zelati, T. M. Elgindi, and K. Widmayer, Enhanced dissipation in the Navier-Stokes equations near the Poiseuille ﬂow, arXiv e-prints (Jan. 2019), arXiv:1901.01571, available at 1901.01571

work page arXiv 2019

[14] [14]

Crippa, R

G. Crippa, R. Luc `a, and C. Schulze,Polynomial mixing under a certain stationary Euler ﬂow, Phys. D 394 (2019), 44–55

work page 2019

[15] [15]

Deng, Resolvent estimates for a two-dimensional non-self-adjoint operator , Commun

W. Deng, Resolvent estimates for a two-dimensional non-self-adjoint operator , Commun. Pure Appl. Anal. 12 (2013), no. 1, 547–596

work page 2013

[16] [16]

Desvillettes and C

L. Desvillettes and C. Villani, On the trend to global equilibrium in spatially inhomogeneous entropy-dissipating systems: the linear Fokker-Planck equation, Comm. Pure Appl. Math. 54 (2001), no. 1, 1–42

work page 2001

[17] [17]

Dolbeault, C

J. Dolbeault, C. Mouhot, and C. Schmeiser, Hypocoercivity for linear kinetic equations conserving mass , Trans. Amer. Math. Soc. 367 (2015), no. 6, 3807–3828

work page 2015

[18] [18]

B Dubrulle and S Nazarenko, On scaling laws for the transition to turbulence in uniform-shear ﬂows , Euro. Phys. Lett. 27 (1994), no. 2, 129

work page 1994

[19] [19]

Gallagher, T

I. Gallagher, T. Gallay, and F. Nier, Spectral asymptotics for large skew-symmetric perturbations of the harmonic oscil- lator, Int. Math. Res. Not. IMRN 12 (2009), 2147–2199

work page 2009

[20] [20]

Gallay, Enhanced dissipation and axisymmetrization of two-dimensional viscous vortices , Arch

T. Gallay, Enhanced dissipation and axisymmetrization of two-dimensional viscous vortices , Arch. Ration. Mech. Anal. 230 (2018), no. 3, 939–975

work page 2018

[21] [21]

M.-H. Giga, Y . Giga, and J. Saal, Nonlinear partial differential equations, Progress in Nonlinear Differential Equations and their Applications, vol. 79, Birkh¨auser Boston, Inc., Boston, MA, 2010

work page 2010

[22] [22]

Linear inviscid damping and enhanced viscous dissipation of shear flows by using the conjugate operator method

E. Grenier, T. T. Nguyen, F. Rousset, and A. Soffer, Linear inviscid damping and enhanced viscous dissipation of shear ﬂows by using the conjugate operator method, ArXiv e-prints (Apr. 2018), available at 1804.08291

work page internal anchor Pith review Pith/arXiv arXiv 2018

[23] [23]

H ´erau, Short and long time behavior of the Fokker-Planck equation in a conﬁning potential and applications, J

F. H ´erau, Short and long time behavior of the Fokker-Planck equation in a conﬁning potential and applications, J. Funct. Anal. 244 (2007), no. 1, 95–118

work page 2007

[24] [24]

H ´erau and F

F. H ´erau and F. Nier,Isotropic hypoellipticity and trend to equilibrium for the Fokker-Planck equation with a high-degree potential, Arch. Ration. Mech. Anal. 171 (2004), no. 2, 151–218. 14 M. COTI ZELATI AND M. DOLCE

work page 2004

[25] [25]

Ibrahim, Y

S. Ibrahim, Y . Maekawa, and N. Masmoudi, On pseudospectral bound for non-selfadjoint operators and its application to stability of Kolmogorov ﬂows, ArXiv e-prints (Oct. 2017), available at 1710.05132

work page arXiv 2017

[26] [26]

Latini and A

M. Latini and A. J. Bernoff, Transient anomalous diffusion in Poiseuille ﬂow , Journal of Fluid Mechanics 441 (2001), 399–411

work page 2001

[27] [27]

T. Li, D. Wei, and Z. Zhang, Pseudospectral and spectral bounds for the Oseen vortices operator , ArXiv e-prints (Jan. 2017), available at 1701.06269

work page internal anchor Pith review Pith/arXiv arXiv 2017

[28] [28]

T. Li, D. Wei, and Z. Zhang, Pseudospectral bound and transition threshold for the 3D Kolmogorov ﬂow, arXiv e-prints (Jan. 2018), available at 1801.05645

work page internal anchor Pith review Pith/arXiv arXiv 2018

[29] [29]

C. J. Miles and C. R. Doering, Diffusion-limited mixing by incompressible ﬂows , Nonlinearity 31 (2018), no. 5, 2346–2350

work page 2018

[30] [30]

P. B. Rhines and W. R. Young, How rapidly is a passive scalar mixed within closed streamlines? , Journal of Fluid Mechanics 133 (1983), 133–145

work page 1983

[31] [31]

Villani, Hypocoercivity, Mem

C. Villani, Hypocoercivity, Mem. Amer. Math. Soc. 202 (2009), no. 950, iv+141

work page 2009

[32] [32]

Transition threshold for the 3D Couette flow in Sobolev space

D. Wei and Z. Zhang, Transition threshold for the 3D Couette ﬂow in Sobolev space, ArXiv e-prints (Mar. 2018), available at 1803.01359

work page internal anchor Pith review Pith/arXiv arXiv 2018

[33] [33]

D. Wei, Z. Zhang, and W. Zhao, Linear inviscid damping and enhanced dissipation for the Kolmogorov ﬂow , ArXiv e-prints (Nov. 2017), available at 1711.01822

work page internal anchor Pith review Pith/arXiv arXiv 2017

[34] [34]

Wei, Diffusion and mixing in ﬂuid ﬂow via the resolvent estimate , arXiv e-prints (Nov

D. Wei, Diffusion and mixing in ﬂuid ﬂow via the resolvent estimate , arXiv e-prints (Nov. 2018), available at 1811. 11904

work page 2018

[35] [35]

Wei and Z

D. Wei and Z. Zhang, Enhanced dissipation for the Kolmogorov ﬂow via the hypocoercivity method, Sci. China Math. 62 (2019), no. 6, 1219–1232. DEPARTMENT OF MATHEMATICS , I MPERIAL COLLEGE LONDON , L ONDON , SW7 2AZ, UK E-mail address: m.coti-zelati@imperial.ac.uk GSSI - G RAN SASSO SCIENCE INSTITUTE , 67100, L’A QUILA , ITALY E-mail address: michele.dolce@gssi.it

work page 2019