  We numerically investigate the evolution of compact high-velocity clouds
(CHVCs) passing through a hot, tenuous gas representing the highly-ionized
circumgalactic medium (CGM) by applying the adaptive-mesh refinement code
FLASH.
  The model clouds start from both hydrostatic and thermal equilibrium and are
in pressure balance with the CGM. Here, we present 14 models, divided into two
mass categories and two metallicities each and different velocities. We allow
for self-gravity and thermal conduction or not. All models experience mass
diffusion, radiative cooling, and external heating leading to dissociation and
ionization.
  Our main findings are: 1) self-gravity stabilizes clouds against
Rayleigh-Taylor instability which are disrupted within 10 sound-crossing times
without; 2) clouds can develop Jeans-instable regions internally even though
they are initially below Jeans mass; 3) all clouds lose mass by ram pressure
and Kelvin-Helmholtz instability; 4) thermal conduction substantially lowers
mass-loss rates, by this, extending the clouds' lifetimes, particularly, more
than doubling the lifetime of low-mass clouds; 5) thermal conduction leads to
continuous, filamentary stripping, while the removed gas is heated up quickly
and mixes efficiently with the ambient CGM; 6) without thermal conduction the
removed gas consists of dense, cool, clumpy fragments; 7) thermal conduction
might prevent CHVCs from forming stars; 8) clouds decelerated by means of drag
from the ambient CGM form head-tail shapes and collapse after they reach
velocities characteristic for intermediate-velocity clouds.
  Conclusively, only sophisticated modelling of CHVCs as non-homogeneous and
non-isothermal clouds with thermal conduction and self-gravity explains
observed morphologies and naturally leads to the suppression of star formation.
