Blow up of smooth highly decreasing at infinity solutions to the   compressible Navier-Stokes equations
  We prove that the smooth solutions to the Cauchy problem for the Navier-Stokes equations with conserved mass, total energy and finite momentum of inertia loses the initial smoothness within a finite time in the case of space of dimension 3 or greater even if the initial data are not compactly supported. 
Olga Rozanova