  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 $L^\infty$-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.
