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Large time behavior and asymptotic stability of the 2D Euler and linearized Euler equations. (English) Zbl 1189.35234
Summary: We study the asymptotic behavior and the asymptotic stability of the 2D Euler equations and of the 2D linearized Euler equations close to parallel flows. We focus on flows with spectrally stable profiles \(U(y)\) and with stationary streamlines \(y=y_{0}\) (such that \(U^{\prime}(y_{0})=0)\), 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.

MSC:
35Q31 Euler equations
76F20 Dynamical systems approach to turbulence
76D05 Navier-Stokes equations for incompressible viscous fluids
76E20 Stability and instability of geophysical and astrophysical flows
76M22 Spectral methods applied to problems in fluid mechanics
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