On the spectral instability of parallel shear flows

Grenier, Emmanuel; Guo, Yan; Nguyen, Toan T.

doi:10.5802/slsedp.82

Cited by 5 publications

(4 citation statements)

References 7 publications

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“…See [29] for a case where linear inviscid damping does not hold for a monotone shear flow. See [39] for a spectral instability result. Nonlinear asymptotic stability for perturbations around a class of monotone shear flows, with compactly supported vorticity in T × [−1, 1], was recently proved by Ionescu-Jia [46] and Masmoudi-Zhao [58].…”

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confidence: 99%

Linear stability analysis for 2D shear flows near Couette in the isentropic Compressible Euler equations

Antonelli,

Dolce,

Marcati

2020

Preprint

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In this paper, we investigate linear stability properties of the 2D isentropic compressible Euler equations linearized around a shear flow given by a monotone profile, close to the Couette flow, with constant density, in the domain T × R. We begin by directly investigating the Couette shear flow, where we characterize the linear growth of the compressible part of the fluid while proving time decay for the incompressible part (inviscid damping with slower rates).Then we extend the analysis to monotone shear flows near Couette, where we are able to give an upper bound, superlinear in time, for the compressible part of the fluid. The incompressible part enjoys an inviscid damping property, analogous to the Couette case. In the pure Couette case, we exploit the presence of an additional conservation law (which connects the vorticity and the density on the moving frame) in order to reduce the number of degrees of freedom of the system. The result then follows by using weighted energy estimates.In the general case, unfortunately, this conservation law no longer holds. Therefore we define a suitable weighted energy functional for the whole system, which can be used to estimate the irrotational component of the velocity but does not provide sharp bounds on the solenoidal component. However, even in the absence of the aforementioned additional conservation law, we are still able to show the existence of a functional relation which allows us to recover somehow the vorticity from the density, on the moving frame. By combining the weighted energy estimates with the functional relation we also recover the inviscid damping for the solenoidal component of the velocity.

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confidence: 99%

Linear stability analysis for 2D shear flows near Couette in the isentropic Compressible Euler equations

Antonelli,

Dolce,

Marcati

2020

Preprint

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“…The reader should also see [2,11,34,54], [40,51,52] for related results. There have also been several works ( [17][18][19][20][21][22]25]) establishing generic invalidity of expansions of the type (1.6) in Sobolev spaces in the unsteady setting.…”

Section: Other Workmentioning

confidence: 99%

Validity of steady Prandtl layer expansions

Guo

Iyer

2023

Comm Pure Appl Math

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Let the viscosity for the 2D steady Navier‐Stokes equations in the region and with no slip boundary conditions at . For , we justify the validity of the steady Prandtl layer expansion for scaled Prandtl layers, including the celebrated Blasius boundary layer. Our uniform estimates in ε are achieved through a fixed‐point scheme: for solving the Navier‐Stokes equations, where are the tangential and normal velocities at , DNS stands for of the vorticity equation for the normal velocity v, and the compatibility ODE for at .

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“…Even for the monotonic and concave Prandtl boundary layer profiles, we may not expect the nonlinear stability of the Prandtl boundary layer in the Sobolev setting. In the notable work [14], the authors studied the linearized Navier--Stokes equations around generic stationary shear flows of the boundary layer type and constructed solutions with highly growing eigenmodes like e t/\nu 1 4 (\nu : viscosity) related to the O(\nu - 3 8 ) tangential frequency; see [13] for related statements and [16,18,19] for new progress. The result in [14] suggests somehow that one can only prove the validity of Prandtl boundary layer theory in the function spaces of Gevrey class, and recently there have been several interesting works in this direction; see [1,9,10].…”

Section: \Left\{mentioning

confidence: 99%

Validity of Prandtl Expansions for Steady MHD in the Sobolev Framework

Liu

Yang

Zhang

2023

SIAM J. Math. Anal.

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\mathrm{ \mathrm{ \mathrm{\mathrm{ \mathrm{ . \mathrm{\mathrm{ \mathrm{\mathrm{. \mathrm{\mathrm{\mathrm{\mathrm{ .© 2023 \mathrm{ \mathrm{\mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{\mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{\mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{\mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{

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On the spectral instability of parallel shear flows

Cited by 5 publications

References 7 publications

Linear stability analysis for 2D shear flows near Couette in the isentropic Compressible Euler equations

Linear stability analysis for 2D shear flows near Couette in the isentropic Compressible Euler equations

Validity of steady Prandtl layer expansions

Validity of Prandtl Expansions for Steady MHD in the Sobolev Framework

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