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

New thought on Matsumura-Nishida theory in the $L_p$-$L_q$ maximalregularity framework