  In this paper, we investigate the Rayleigh-Taylor instability problem for two
compressible, immiscible, inviscid flows rotating with an constant angular
velocity, and evolving with a free interface in the presence of a uniform
gravitational field. First we construct the Rayleigh-Taylor steady-state
solutions with a denser fluid lying above the free interface with the second
fluid, then we turn to an analysis of the equations obtained from linearization
around such a steady state. In the presence of uniform rotation, there is no
natural variational framework for constructing growing mode solutions to the
linearized problem. Using the general method of studying a family of modified
variational problems introduced in \cite{Y-I2}, we construct normal mode
solutions that grow exponentially in time with rate like
$e^{t\sqrt{c|\xi|-1}}$, where $\xi$ is the spatial frequency of the normal mode
and the constant $c$ depends on some physical parameters of the two layer
fluids. A Fourier synthesis of these normal mode solutions allows us to
construct solutions that grow arbitrarily quickly in the Sobolev space $H^k$,
and lead to an ill-posedness result for the linearized problem. Moreover, from
the analysis we see that rotation diminishes the growth of instability. Using
the pathological solutions, we then demonstrate the ill-posedness for the
original non-linear problem in some sense.
