We consider the incompressible Euler equations in R 2 when the initial vorticity is bounded, radially symmetric and non-increasing in the radial direction. Such a radial distribution is stationary, and we show that the monotonicity produces stability in some weighted norm related to the angular impulse. For instance, it covers the cases of circular vortex patches and Gaussian distributions. Our stability does not depend on L ∞ -bound or support size of perturbations. The proof is based on the fact that such a radial monotone distribution minimizes the impulse of functions having the same level set measure.

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We consider the incompressible Euler equations in the half cylinder R>0 × T. In this domain, any vorticity which is independent of x2 defines a stationary solution. We prove that such a stationary solution is nonlinearly stable in a weighted L 1 norm involving the horizontal impulse, if the vorticity is non-negative and non-increasing in x1. This includes stability of cylindrical patches {x1 < α}, α > 0. The stability result is based on the fact that such a profile is the unique minimizer of the horizontal impulse among all functions with the same distribution function. Based on stability, we prove existence of vortex patches in the half cylinder that exhibit infinite perimeter growth in infinite time.