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

Abstract

Let $L_m^2({\mathbb{R}}^2)$ be the space of square-integrable functions on ${\mathbb{R}}^2$ with $m$-fold rotational symmetry. Let $\overline{\omega }(r,\theta )=r^{-\alpha }f(\theta )$ be a homogeneous steady state of the two-dimensional incompressible Euler equations with $(m\cdot l)$-fold rotational symmetry. If $f(\theta )$ is a constant function we say that $\overline{\omega }$ is a radial power-law vortex. We prove that the incompressible Euler equations in vorticity form, linearized around any homogeneous steady state $\overline{\omega }$ that is not a radial power-law vortex, are ill-posed on $L_m^2({\mathbb{R}}^2)$ for any $m\ge 2$, any $l\ge 1$ and $\alpha \in (0,1)$.

13 pages