circular flows with power law $U(r)\sim r^{\alpha}, \alpha \in \R$ as $r\downarrow 0$ and $U(r)\sim r^{\beta}, \beta\in \R$ as $r\uparrow \infty$ as well as degenerate shear flows like $U(y)=\e^{y}$ in a periodic channel $\T
\times I$. As the main stability assumption, here we do not require strict monotonicity, but rather require that $$\begin{aligned}
   \sign(U') U''' \leq c |U'|,\end{aligned}$$ which is natural in view of explicit and scaling results for polynomial and exponential shear flow profiles on bounded intervals. Due to technical obstructions and the locally perturbative nature of our method, some further conditions are required on the underlying flow (c.f. Section \[sec:mildly\_degen\]) We remark that this condition is different from Arnold’s stability criterion, which is given by a control of $U''/U$ from above and below.

Stability of radially symmetric, strictly monotone decreasing distributions of vorticity was studied recently in [@BCZVvortex2017], using a spectral approach, devising interesting phenomena such as *vortex axysimmetrization* *vorticity depletion* (see [@Euler_stability] and [@WZZvorticitydepl]). Our work can be seen as complementary to this one: we do not require monotonicity of the profile and we can handle vorticity profiles that blow up at $r=0$.

In the following we recall the linearized Euler equations for shear flows and circular 