On the Global Motion of Viscous Compressible Barotropic Flows Subject to Large External Potential Forces and Vacuum

Abstract

We study the Cauchy problem of three-dimensional compressible barotropic Navier--Stokes equations with large external potential forces. We first prove that the Cauchy problem has a unique global strong solution with large oscillations and interior vacuum, provided the initial data are of small energy and the unique steady state is strictly away from vacuum. Next, we show that if the initial data are more regular and satisfy some additional compatibility conditions, the strong solution is indeed a classical one away from the initial time. Finally, as by-products, the global existence and large-time behavior of weak solutions with large external potential forces and discontinuous initial data containing vacuum are also obtained.