On Blowup of Classical Solutions to the Compressible Navier-Stokes Equations

In this paper, we study the finite time blow up of smooth solutions to the Compressible Navier-Stokes system when the initial data contain vacuums. We prove that any classical solutions of viscous compressible fluids without heat conduction will blow up in finite time, as long as the initial data has an isolated mass group (see Definition 2.2). The results hold regardless of either the size of the initial data or the far fields being vacuum or not. This improves the blowup results of Xin (Comm Pure Appl Math 51:229–240, 1998) by removing the crucial assumptions that the initial density has compact support and the smooth solution has finite total energy. Furthermore, the analysis here also yields that any classical solutions of viscous compressible fluids without heat conduction in bounded domains or periodic domains will blow up in finite time, if the initial data have an isolated mass group satisfying some suitable conditions.