Аннотация: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.