Flexibility and Rigidity in Steady Fluid Motion

Constantin, Peter; Drivas, Theodore D.; Ginsberg, Daniel

doi:10.1007/s00220-021-04048-4

Cited by 37 publications

(39 citation statements)

References 16 publications

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“…Note that the number −6 is the second eigenvalue of the Laplace-Beltrami operator. Such symmetry results are proved in Constantin-Drivas-Ginsberg [15] for general Riemannian surfaces with Killing field, under the condition that F is larger than the smallest eigenvalue of the Laplace-Beltrami operator, which on the sphere amounts to F > −2. The symmetric structure of the sphere explains the improvement that we are able to obtain.…”

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confidence: 79%

Stratospheric planetary flows from the perspective of the Euler equation on a rotating sphere

Constantin,

Germain

2021

Preprint

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This article is devoted to stationary solutions of Euler's equation on a rotating sphere, and to their relevance to the dynamics of stratospheric flows in the atmosphere of the outer planets of our solar system and in polar regions of the Earth. For the Euler equation, under appropriate conditions, rigidity results are established, ensuring that the solutions are either zonal or rotated zonal solutions. A natural analogue of Arnold's stability criterion is proved. In both cases, the lowest mode Rossby-Haurwitz stationary solutions (more precisely, those whose stream functions belong to the sum of the first two eigenspaces of the Laplace-Beltrami operator) appear as limiting cases. We study the stability properties of these critical stationary solutions. Results on the local and global bifurcation of non-zonal stationary solutions from classical Rossby-Haurwitz waves are also obtained. Finally, we show that stationary solutions of the Euler equation on a rotating sphere are building blocks for travelling-wave solutions of the 3D system that describes the leading order dynamics of stratospheric planetary flows, capturing the characteristic decrease of density and increase of temperature with height in this region of the atmosphere.

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confidence: 79%

Stratospheric planetary flows from the perspective of the Euler equation on a rotating sphere

Constantin,

Germain

2021

Preprint

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“…This rigidity confers the idea that certain steady solutions adapt to the geometry and symmetries of the domain they occupy. Further results in this direction are given in [13,17].…”

Section: Euler Equations On a 2d Euclidean Domainmentioning

confidence: 91%

On zonal steady solutions to the 2D Euler equations on the rotating unit sphere

Nualart¹

2022

Preprint

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The present paper studies the structure of the set of stationary solutions to the incompressible Euler equations on the rotating unit sphere that are near two basic zonal flows: the zonal Rossby-Haurwitz solution of degree 2 and the zonal rigid rotation Y 0 1 along the polar axis. We construct a new family of non-zonal steady solutions arbitrarily close in analytic regularity to the second degree zonal Rossby-Haurwitz stream function, for any given rotation of the sphere. This shows that any non-linear inviscid damping to a zonal flow cannot be expected for solutions near this Rossby-Haurwitz solution.On the other hand, we prove that, under suitable conditions on the rotation of the sphere, any stationary solution close enough to the rigid rotation zonal flow Y 0 1 must itself be zonal, witnessing some sort of rigidity inherited from the equation, the geometry of the sphere and the base flow. Nevertheless, when the conditions on the rotation of the sphere fail, we are able to prove the existence of explicit stationary non-zonal solutions bifurcating from Y 0 1 , in the same spirit as those emanating from the zonal Rossby-Haurwitz solution of degree 2.

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“…This type of rigidity question has been very lately understood for different equations and different settings such as in the papers by Koch-Nadirashvili-Sverak [20] for Navier-Stokes, Hamel-Nadirashvili [16][17][18] for the 2D Euler equation on a strip, punctured disk or the full plane, Gómez-Serrano-Park-Shi-Yao [14] for the 2D Euler and modified SQG in the full plane and Constantin-Drivas-Ginsberg [8] for the 2D and 3D Euler, as well as the 2D Boussinesq and the 3D Magnetohydrostatic (MHS) equations.…”

Section: ω(X T) =mentioning

confidence: 99%

Remarks on Stationary and Uniformly-rotating Vortex Sheets: Rigidity Results

Gómez-Serrano

Park

Shi

et al. 2021

Commun. Math. Phys.

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In this paper, we show that the only solution of the vortex sheet equation, either stationary or uniformly rotating with negative angular velocity $$\Omega $$ Ω , such that it has positive vorticity and is concentrated in a finite disjoint union of smooth curves with finite length is the trivial one: constant vorticity amplitude supported on a union of nested, concentric circles. The proof follows a desingularization argument and a calculus of variations flavor.

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Flexibility and Rigidity in Steady Fluid Motion

Cited by 37 publications

References 16 publications

Stratospheric planetary flows from the perspective of the Euler equation on a rotating sphere

Stratospheric planetary flows from the perspective of the Euler equation on a rotating sphere

On zonal steady solutions to the 2D Euler equations on the rotating unit sphere

Remarks on Stationary and Uniformly-rotating Vortex Sheets: Rigidity Results

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