  We study the stability of the vortex in a 2D model of continuous compressible
media in a uniformly rotating reference frame. As it is known, the axisymmetric
vortex in a fixed reference frame is stable with respect to asymmetric
perturbations for the solution of the 2D incompressible Euler equations and
basically instable for compressible Euler equations. We show that the situation
is quite different for a compressible axisymmetric vortex in a rotating
reference frame. First, we consider special solutions with linear profile of
velocity (or with spatially-uniform velocity gradients), which are important
because many real vortices have similar structure near their centers. We
analyze both cyclonic and anticyclonic cases and show that the stability of the
solution depends only on the ratio of the vorticity to the Coriolis parameter.
Using a very delicate analysis along with computer aided proof, we show that
the stability of solutions can take place only for a narrow range of this
ratio. Our results imply that the rotation of the coordinate frame can
stabilize the compressible vortex. Further, we perform both analytical and
numerical analysis of stability for real-shaped vortices.
