Linear inviscid damping and vorticity depletion for shear flows

Wei, Dongyi; Zhang, Zhifei; Wei, Zhao

doi:10.48550/arxiv.1704.00428

Cited by 22 publications

(36 citation statements)

References 9 publications

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“…Landau damping is considered fundamental to understanding the dynamics of collisionless and weakly collisional plasmas by the physics community [102,26] and it has been speculated that inviscid damping should play a related role in understanding the dynamics of the 2D Euler equations [6,55,88,124,103], with applications to cyclotron dynamics [31] and atmospheric sciences [87,106,118]. See [79,119,129,131,121,78,123,120,21] for mathematical works on inviscid damping in the linearized 2D Euler equations.…”

Section: Hydrodynamic Stability At High Reynolds Numbermentioning

confidence: 99%

“…Linear works studying enhanced dissipation and inviscid damping in 2D Navier-Stokes and Euler have been quite technical and the theory is still in its infancy (see e.g. [130,132,21,119,121,53,75,120,67]). We are not aware of any analogous works for 3D Navier-Stokes or 3D Euler, indeed, this kind of detailed information about the linearized operators in 3D seems to be completely open (except for Couette flow) -even for the original problem of Reynolds [99,40] after 120 years of research.…”

Section: Open Problemsmentioning

confidence: 99%

See 1 more Smart Citation

Stability of the Couette flow at high Reynolds numbers in two dimensions and three dimensions

Bedrossian¹,

Germain

Masmoudi

2018

Bull. Amer. Math. Soc.

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We review works on the asymptotic stability of the Couette flow. The majority of the paper is aimed towards a wide range of applied mathematicians with an additional section aimed towards experts in the mathematical analysis of PDEs. Contents Hydrodynamic stability at high Reynolds number1 2. Linear dynamics in dimension d = 2 8 3. Nonlinear dynamics in dimension d = 2 11 4. Linear dynamics in dimension d = 3 1 As remarked above, in order to get non-trivial equilibria, we usually need to impose boundary conditions or an external force field. However, let us ignore this for the moment, as it is not directly relevant to our discussions. STABILITY OF THE COUETTE FLOW AT HIGH REYNOLDS NUMBERS IN 2D AND 3D3 simply by dropping the quadratic nonlinearity in (1.1):( 1.3)The simplest notion of linear stability is spectral stability:Many classical theories of hydrodynamic stability are focused on studies of spectral stability (see [122] and the references therein). In the context of shear flows at high Reynolds number, the most famous results are Rayleigh's inflection point theorem [96,40] and its refinement due to Fjørtoft [48], regarding the stability of shear flows in the 2D Euler equations (ν = 0). These results (unfortunately) extend to planar 3D shear flows via Squire's theorem [107,39,40]. Squire's theorem for inviscid flows states: If the 3D Euler equations linearized around the shear flow u E = (f (y), 0, 0) have unstable eigenvalues, then so do the 2D Euler equations linearized around u E = (f (y), 0) (the converse implication being obvious). Therefore, for any shear flow of this type, spectral stability in 2D implies spectral stability in 3D. This is 'unfortunate' because it gives the false impression that most of the interesting aspects of hydrodynamic stability can be found in the 2D equations. As we will see, hydrodynamic stability in 2D and 3D, at high Reynolds numbers, are quite distinct.Spectral stability is not the only relevant definition of linear stability. In particular, we have the following distinct definition, which was suggested as more natural in fluid mechanics by Kelvin [68], Orr [92], and later Case [29]and Dikii [38].Definition 1.2 (Lyapunov linear stability). Given two norms X and Y (often assumed the same), the equilibrium u E is called linearly stable (from X to Y ) if

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Section: Hydrodynamic Stability At High Reynolds Numbermentioning

confidence: 99%

Section: Open Problemsmentioning

confidence: 99%

Stability of the Couette flow at high Reynolds numbers in two dimensions and three dimensions

Bedrossian¹,

Germain

Masmoudi

2018

Bull. Amer. Math. Soc.

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“…This re-writes the evolution entirely on the density as a Volterra equation -only a slight generalization of an ODE, so in a sense, we have essentially explicitly diagonalized (11). Such a structure is unfortunately not present in the related fluid mechanics problems [56,52,53,4] which makes those linear problems much more difficult. However, the structure does survive (barely) when adding certain collision operators [49,3].…”

Section: Linearized Vlasov Equationsmentioning

confidence: 99%

A brief summary of nonlinear echoes and Landau damping

Bedrossian¹

2018

Journées Équations Aux Dérivées Partielles

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In this expository note we review some recent results on Landau damping in the nonlinear Vlasov equations, focusing specifically on the recent construction of nonlinear echo solutions by the author [arXiv:1605.06841] and the associated background. These solutions show that a straightforward extension of Mouhot and Villani's theorem on Landau damping to Sobolev spaces on T n x × R n v is impossible and hence emphasize the subtle dependence on regularity of phase mixing problems. This expository note is specifically aimed at mathematicians who study the analysis of PDEs, but not necessarily those who work specifically on kinetic theory. However, for the sake of brevity, this review is certainly not comprehensive.

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“…When β = 0, the nonexistence of nonzero imaginary eigenvalues and, as a consequence, the linear damping is true for flows in class K + (see [22]). Recently, when β = 0, under the assumption that the linearized operator has no embedding eigenvalues, more explicit linear decay estimates of the velocity were obtained for symmetric and monotone shear flows in [39,40,45] with more regular initial data (e.g. ω (0) ∈ H 1 or H 2 ).…”

Section: Linear Inviscid Dampingmentioning

confidence: 99%

Barotropic instability of shear flows

2020

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We consider barotropic instability of shear flows for incompressible fluids with Coriolis effects. For a class of shear flows, we develop a new method to find the sharp stability conditions. We study the flow with Sinus profile in details and obtain the sharp stability boundary in the whole parameter space, which corrects previous results in the fluid literature. Our new results are confirmed by more accurate numerical computation. The addition of the Coriolis force is found to bring fundamental changes to the stability of shear flows. Moreover, we study dynamical behaviors near the shear flows, including the bifurcation of nontrivial traveling wave solutions and the linear inviscid damping. The first ingredient of our proof is a careful classification of the neutral modes. The second one is to write the linearized fluid equation in a Hamiltonian form and then use an instability index theory for general Hamiltonian partial differential equations. The last one is to study the singular and nonresonant neutral modes using Sturm‐Liouville theory and hypergeometric functions.

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Linear inviscid damping and vorticity depletion for shear flows

Cited by 22 publications

References 9 publications

Stability of the Couette flow at high Reynolds numbers in two dimensions and three dimensions

Stability of the Couette flow at high Reynolds numbers in two dimensions and three dimensions

A brief summary of nonlinear echoes and Landau damping

Barotropic instability of shear flows

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