  We are concerned with an isothermal model of viscous and capillary
compressible fluids derived by J. E. Dunn and J. Serrin (1985), which can be
used as a phase transition model. Compared with the classical compressible
Navier-Stokes equations, there is a smoothing effect on the density that comes
from the capillary terms. First, we prove that the global solutions with
critical regularity that have been constructed in [11] by the second author and
B. Desjardins (2001), are Gevrey analytic. Second, we extend that result to a
more general critical L p framework. As a consequence, we obtain algebraic
time-decay estimates in critical Besov spaces (and even exponential decay for
the high frequencies) for any derivatives of the solution. Our approach is
partly inspired by the work of Bae, Biswas \& Tadmor [2] dedicated to the
classical incompressible Navier-Stokes equations, and requires our establishing
new bilinear estimates (of independent interest) involving the Gevrey
regularity for the product or composition of functions. To the best of our
knowledge, this is the first work pointing out Gevrey analyticity for a model
of compressible fluids.
