  In this paper, we prove the global wellposedness of the Navier-Stokes
equations describing a motion of compressible, viscous, barotropic fluid flow
in a 3 dim. exterior domain in the $L_p$ in time and $L_2 \cap L_6$ maximal
regularity framework. This is an extension of a famous thoerem due to
Matsumura-Nishida Commun Math. Phys. 89 (1983), 445--464. In Matsumura and
Nishida theory, they used energy method and their requirement was that space
derivatives of the mass density up to third order and space derivatives of the
velocity fields up to fourth order belong to $L_2$ in space-time. On the other
hand, in the present manuscript space derivatives of the mass density up to
first order and the space derivatives of the velocity fields up to second order
belong to $L_2$ in maximal and $L_2 \cap L_6$ in space. The proof is based on
the $L_p$-$L_q$ maximal regularity and decay properties of solutions to the
linearized equations, namely Stokes equations appering in the study of
compressible fluid flows.
