Global solutions of the Navier–Stokes equations for viscous compressible flows

Abstract

The compressible Navier–Stokes equations for viscous flows with general large continuous initial data, as well as with large discontinuous initial data, are studied. Both a homogeneous free boundary problem with zero outer pressure and a fixed boundary problem are considered. For the large initial data in H 1, the existence, uniqueness, and regularity of global solutions in H 1 for real viscous flows are established, and it is showed that neither shock waves nor vacuum and concentration in the solutions are developed in a finite time. For the large discontinuous data, the global existence of large weak solutions for the perfect gases is also established using a different argument, and it is indicated that the solutions do not develop vacuum or concentration although the solutions have large discontinuity. For the free boundary problem, the interface separating the flows from the zero outer pressure expands at a finite speed.