Linear Inviscid Damping in Gevrey Spaces

Jia, Hao

doi:10.1007/s00205-019-01445-x

Cited by 43 publications

(40 citation statements)

References 27 publications

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“…We further stress that, while stability of the linearized problem in Sobolev [12,14,15] and Gevrey spaces [8] is fundamental to attack the nonlinear problem, this article shows that it is further essential to understand the linearization around non-shear low frequency perturbations, which appear naturally in the nonlinear problem.…”

Section: Discussionmentioning

confidence: 96%

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Echo Chains as a Linear Mechanism: Norm Inflation, Modified Exponents and Asymptotics

Deng

Zillinger

2021

Arch Rational Mech Anal

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In this article we show that the Euler equations, when linearized around a low frequency perturbation to Couette flow, exhibit norm inflation in Gevrey-type spaces as time tends to infinity. Thus, echo chains are shown to be a (secondary) linear instability mechanism. Furthermore, we develop a more precise analysis of cancellations in the resonance mechanism, which yields a modified exponent in the high frequency regime. This allows us, in addition, to remove a logarithmic constraint on the perturbations present in prior works by Bedrossian, Deng and Masmoudi, and to construct solutions which are initially in a Gevrey class for which the velocity asymptotically converges in Sobolev regularity but diverges in Gevrey regularity.

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Section: Discussionmentioning

confidence: 96%

“…While sufficient to establish linear inviscid damping with the optimal decay rates of the velocity perturbation, this leaves a large gap compared to the Gevrey regularity requirement of existing nonlinear results. Indeed, in [7,8] Ionescu and Jia further impose a compact support assumption to establish linear stability in Gevrey spaces.…”

Section: Introductionmentioning

confidence: 99%

Echo Chains as a Linear Mechanism: Norm Inflation, Modified Exponents and Asymptotics

Deng

Zillinger

2021

Arch Rational Mech Anal

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“…For simplicity of calculation and presentation we neglect the effects of boundaries and consider the setting of an infinite periodic channel T L × R. We however expect that our results should extend with moderate technical effort to the setting of smooth compactly supported perturbations to Couette flow (which is considered in [Jia19]) and to more general Bilipschitz flows as considered in [Zil16].…”

Section: Introductionmentioning

confidence: 89%

On the smallness condition in linear inviscid damping: monotonicity and resonance chains

Deng¹,

Zillinger

2020

Nonlinearity

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We consider the effects of mixing by smooth bilipschitz shear flows in the linearized Euler equations on T L × R. Here, we construct a model which is closely related to a small high frequency perturbation around Couette flow, which exhibits linear inviscid damping for L sufficiently small, but for which damping fails if L is large. In particular, similar to the instability results for convex profiles for a shear flow being bilipschitz is not sufficient for linear inviscid damping to hold. Instead of an eigenvalue-based argument the underlying mechanism here is shown to be based on a new cascade of resonances moving to higher and higher frequencies in y, which is distinct from the echo chain mechanism in the nonlinear problem.

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“…It is not clear at the moment whether the linear decay estimates can be applied to the nonlinear analysis. 1 1 See, however, the recent result [22], which established linear inviscid damping in Gevrey spaces and appears to be more promising for nonlinear applications. The methods introduced in this paper played an important role in [22].…”

Section: The General Inviscid Damping Problemmentioning

confidence: 99%

Axi‐symmetrization near Point Vortex Solutions for the 2D Euler Equation

Ionescu¹,

Jia

2021

Comm Pure Appl Math

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We prove asymptotic stability of point vortex solutions to the full Euler equation in two dimensions. More precisely, we show that a small, Gevrey smooth, and compactly supported perturbation of a point vortex leads to a global solution of the Euler equation in 2D, which converges weakly as t ! 1 to a radial profile with respect to the vortex. The position of the point vortex, which is time dependent, stabilizes rapidly and becomes the center of the final, radial profile. The mechanism that leads to stabilization is mixing and inviscid damping.

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Linear Inviscid Damping in Gevrey Spaces

Cited by 43 publications

References 27 publications

Echo Chains as a Linear Mechanism: Norm Inflation, Modified Exponents and Asymptotics

Echo Chains as a Linear Mechanism: Norm Inflation, Modified Exponents and Asymptotics

On the smallness condition in linear inviscid damping: monotonicity and resonance chains

Axi‐symmetrization near Point Vortex Solutions for the 2D Euler Equation

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