Ill-Posedness of the Euler Equations Linearized around Homogeneous Steady States

Let $L^2_m(\mathbb{R}^2)$ be the space of square-integrable functions on $\mathbb{R}^2$ with $m$-fold rotational symmetry. Let $\overlineω(r,θ) = r^{-α}f(θ)$ be a homogeneous steady state of the two-dimensional incompressible Euler equations with $(m\cdot l)$-fold rotational symmetry. If $f(θ)$ is a