Traveling Waves Near Couette Flow for the 2D Euler Equation

Castro, Ángel; Lear, Daniel

doi:10.1007/s00220-023-04636-6

Cited by 6 publications

(2 citation statements)

References 102 publications

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“…Indeed, one could take as an initial condition the non-shear stationary solution. Results on this geometrical property are available for some basic shear flows and highlight the role of the regularity: Lin and Zeng answered the rigidity/flexibility dichotomy for the Couette flow in the periodic channel in [28], Castro and Lear proved similar results for Couette in a periodic strip in [8] and partial answers for the Poiseuille and Kolmogorov flows were obtained by Coti Zelati et al [14].…”

Section: Perspectivesmentioning

confidence: 84%

“…They also constructed non-shear steady solutions whose vorticity is arbitrarily close to the Couette flow vorticity in H s , with s < 3/2, implying that inviscid damping to a shear flow is not true in low regularity. In [8], the authors obtained a new family of non-trivial (non-shear) and smooth travelling waves for the 2D Euler equation in a periodic strip, with the associated vorticity being arbitrarily close to Couette in H s , with s < 3/2, also denying the possibility of non-linear inviscid damping back to a shear flow.…”

Section: Perspectivesmentioning

confidence: 99%

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

Nualart

2023

Nonlinearity

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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 1 0 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 1 0 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, the set of solutions is much richer and we are able to prove the existence of both explicit stationary and travelling wave non-zonal solutions bifurcating from Y 1 0 , in the same spirit as those emanating from the zonal Rossby–Haurwitz solution of degree 2.

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Section: Perspectivesmentioning

confidence: 84%

Section: Perspectivesmentioning

confidence: 99%

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

Nualart

2023

Nonlinearity

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Asymptotic Stability of Couette Flow in a Strong Uniform Magnetic Field for the Euler-MHD System

Zhao,

2024

Arch Rational Mech Anal

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Space Quasi-Periodic Steady Euler Flows Close to the Inviscid Couette Flow

Franzoi,

Masmoudi,

Montalto

2024

Arch Rational Mech Anal

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We prove the existence of steady space quasi-periodic stream functions, solutions for the Euler equation in a vorticity-stream function formulation in the two dimensional channel $${{\mathbb {R}}}\times [-1,1]$$ R × [ - 1 , 1 ] . These solutions bifurcate from a prescribed shear equilibrium near the Couette flow, whose profile induces finitely many modes of oscillations in the horizontal direction for the linearized problem. Using a Nash–Moser implicit function iterative scheme, near such equilibrium we construct small amplitude, space reversible stream functions, slightly deforming the linear solutions and retaining the horizontal quasi-periodic structure. These solutions exist for most values of the parameters characterizing the shear equilibrium. As a by-product, the streamlines of the nonlinear flow exhibit Kelvin’s cat eye-like trajectories arising from the finitely many stagnation lines of the shear equilibrium.

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Traveling Waves Near Couette Flow for the 2D Euler Equation

Cited by 6 publications

References 102 publications

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

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

Asymptotic Stability of Couette Flow in a Strong Uniform Magnetic Field for the Euler-MHD System

Space Quasi-Periodic Steady Euler Flows Close to the Inviscid Couette Flow

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