On recent developments in the spectral problem for the linearized Euler equation

Shvydkoy, Roman; Friedlander, Susan

doi:10.1090/conm/371/06860

Cited by 12 publications

(11 citation statements)

References 34 publications

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“…The result is the operator of the same form as P t , only with B t replaced by the tensor product of B t with m copies of the fundamental matrix of the ξ-equation (51b). For the Euler equation this construction is explained in [18]. Repeating the same argument as in the proof of Theorem 3.1 we arrive at the following formula…”

Section: The Spectrum Of the Sqg Equationmentioning

confidence: 88%

“…Theorems 3.1 and 3.2, which detail the structure of the unstable SQG spectra and imply the spectral mapping property, also hold for the vorticity form of the 2D Euler equation with the scalar vorticity ω substituted for the temperature θ. A summary of the spectral properties of the 2D Euler equation and the equality of the spectra of the velocity and vorticity operators on the respective spaces of the Sobolev tower can be found in Shvydkoy and Friedlander [18]. We remark that to date we have far less information about the spectra of the 3D Euler equation which is much less amenable to the tools that have been used successfully to characterize the spectra of the 2D Euler and the SQG equations.…”

Section: Discussionmentioning

confidence: 99%

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The Unstable Spectrum of the Surface Quasi-Geostrophic Equation

Friedlander

Shvydkoy

2005

J. math. fluid mech.

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Abstract. We study the unstable spectrum of an equation that arises in geophysical fluid dynamics known as the surface quasi-geostrophic equation. In general the spectrum is the union of discrete eigenvalues and an essential spectrum. We demonstrate the existence of unstable eigenvalues in a particular example. We examine the spectra of the semigroup and the evolution operator. We exhibit the structure of these spectra for general flows and prove that a spectral mapping theorem holds. We observe that the spectral properties of the SQG equation are closely analogous to those of the two dimensional Euler equation .

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Section: The Spectrum Of the Sqg Equationmentioning

confidence: 88%

Section: Discussionmentioning

confidence: 99%

The Unstable Spectrum of the Surface Quasi-Geostrophic Equation

Friedlander

Shvydkoy

2005

J. math. fluid mech.

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“…We note that there is no contribution of order 1/t in this case, because G k,y (y, y ′ ) has a discontinuity of its first derivative only, for y ′ = y (see (60) in appendix B ; moreover in the case of a bounded domain G k vanishes at the boundaries (G k,y (y, L) = 0)).…”

Section: A-2) Oscillating Integrals and The Velocity Asymptotic Expanmentioning

confidence: 95%

Large time behavior and asymptotic stability of the 2D Euler and linearized Euler equations

Bouchet

Morita

2010

Physica D: Nonlinear Phenomena

135

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We study the asymptotic behavior and the asymptotic stability of the two-dimensional Euler equations and of the two-dimensional linearized Euler equations close to parallel flows. We focus on flows with spectrally stable profiles U (y) and with stationary streamlines y = y0 (such that U ′ (y0) = 0), a case that has not been studied previously. We describe a new dynamical phenomenon: the depletion of the vorticity at the stationary streamlines. An unexpected consequence, is that the velocity decays for large times with power laws, similarly to what happens in the case of the Orr mechanism for base flows without stationary streamlines. The asymptotic behaviors of velocity and the asymptotic profiles of vorticity are theoretically predicted and compared with direct numerical simulations. We argue on the asymptotic stability of this ensemble of flow profiles even in the absence of any dissipative mechanisms.

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“…In this case we consider the dynamical spectrum of the cocycle restricted to the submanifold u(x) · ξ = 0. This condition has already been used in [20] and its necessity was indicated in [39].…”

Section: General Inclusion Theoremmentioning

confidence: 99%

The Essential Spectrum of Advective Equations

Shvydkoy

2006

Commun. Math. Phys.

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The geometric optics stability method is extended to a general class of linear advective PDE's with pseudodifferential bounded perturbation. We give a new short proof of Vishik's formula for the essential spectral radius. We show that every point in the dynamical spectrum of the corresponding bicharacteristicamplitude system contributes a point into the essential spectrum of the PDE. Generic spectral pictures are obtained in Sobolev spaces of sufficiently large smoothness. Applications to instability are presented.

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On recent developments in the spectral problem for the linearized Euler equation

Cited by 12 publications

References 34 publications

The Unstable Spectrum of the Surface Quasi-Geostrophic Equation

The Unstable Spectrum of the Surface Quasi-Geostrophic Equation

Large time behavior and asymptotic stability of the 2D Euler and linearized Euler equations

The Essential Spectrum of Advective Equations

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