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We study the behavior of solutions to the incompressible $2d$ Euler equations near two canonical shear flows with critical points, the Kolmogorov and Poiseuille flows, with consequences for the associated Navier-Stokes problems. We exhibit a large family of new, non-trivial stationary states of analytic regularity, that are arbitrarily close to the Kolmogorov flow on the square torus $mathbb{T}^2$. This situation contrasts strongly with the setting of some monotone shear flows, such as the Couette flow: in both cases the linearized problem exhibits an inviscid damping mechanism that leads to relaxation of perturbations of the base flows back to nearby shear flows. While this effect persists nonlinearly for suitably small and regular perturbations of some monotone shear flows, for the Kolmogorov flow our result shows that this is not possible. Our construction of these stationary states builds on a degeneracy in the global structure of the Kolmogorov flow on $mathbb{T}^2$. In this regard both the Kolmogorov flow on a rectangular torus and the Poiseuille flow in a channel are very different, and we show that the only stationary states near them must indeed be shears, even in relatively low regularity $H^3$ resp. $H^{5+}$. In addition, we show that this behavior is mirrored closely in the related Navier-Stokes settings: the linearized problems near the Poiseuille and Kolmogorov flows both exhibit an enhanced rate of dissipation. Previous work by us and others shows that this effect survives in the full, nonlinear problem near the Poiseuille flow and near the Kolmogorov flow on rectangular tori, provided that the perturbations lie below a certain threshold. However, we show here that the corresponding result cannot hold near the Kolmogorov flow on $mathbb{T}^2$.
We consider solutions to the 2d Navier-Stokes equations on $mathbb{T}timesmathbb{R}$ close to the Poiseuille flow, with small viscosity $ u>0$. Our first result concerns a semigroup estimate for the linearized problem. Here we show that the $x$-depen
We investigate a steady planar flow of an ideal fluid in a (bounded or unbounded) domain $Omegasubset mathbb{R}^2$. Let $kappa_i ot=0$, $i=1,ldots, m$, be $m$ arbitrary fixed constants. For any given non-degenerate critical point $mathbf{x}_0=(x_{0,1
We derive analogues of the classical Rayleigh, Fjortoft and Arnold stability and instability theorems in the context of the 2D $alpha$-Euler equations.
In this paper, we consider steady Euler flows in two-dimensional bounded annuli, as well as in exterior circular domains, in punctured disks and in the punctured plane. We always assume rigid wall boundary conditions. We prove that, if the flow does
This note is devoted to investigating Liouville type properties of the two dimensional stationary incompressible Magnetohydrodynamics equations. More precisely, under smallness conditions only on the magnetic field, we show that there are no non-triv