Linear Inviscid Damping for a Class of Monotone Shear Flow in Sobolev Spaces

Wei, Dongyi; Zhang, Zhifei; Wei, Zhao

doi:10.1002/cpa.21672

Cited by 105 publications

(136 citation statements)

References 27 publications

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“…

We give an elementary proof of sharp decay rates and the linear inviscid damping near monotone shear flow in a periodic channel, first obtained in [14]. We shall also obtain the precise asymptotics of the solutions, measured in the space L ∞ .

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mentioning

confidence: 88%

“…Zillinger [17,18] and references therein for shear flows close to Couette. In an important work, Wei, Zhang and Zhao in [14] obtained the optimal decay estimates for the linearized problem around monotone shear flows, under very general conditions. We also refer the reader to important developments for the linear inviscid damping in the case of non-monotone shear flows [15,16] and circular flows [2,19].…”

Section: Introductionmentioning

confidence: 99%

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Linear Inviscid Damping Near Monotone Shear Flows

Jia¹

2020

SIAM J. Math. Anal.

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“…

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confidence: 88%

Section: Introductionmentioning

confidence: 99%

Linear Inviscid Damping Near Monotone Shear Flows

Jia¹

2020

SIAM J. Math. Anal.

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“…Explicit decay estimates of the velocity were obtained for monotone and symmetric flows in Refs. with more regular initial data (eg,

ω false(0 false) \in H^{1}

or H 2 ). For

β \neq 0

, linear damping was shown for a class of flows, and explicit decay estimates of the velocity were obtained for monotone flows in Ref.…”

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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“…It is therefore not surprising that a way to quantify this is through the decay of a negative Sobolev norm, as in (1.12). In the context of passive scalars, this point of view was introduced in [31], and it is deeply connected with the regularity of transport equations [12,18,26,45], the quantification and lower bounds on mixing rates [1,6,19,25,32,36,46], and the inviscid damping in the two-dimensional Euler equations linearized around shear flows [8,17,23,40,42,43,[47][48][49].…”

Section: Introductionmentioning

confidence: 99%