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

Abstract

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.