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 total mass, finite total energy and finite momentum of inertia lose 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. The cases of isentropic and incompressible fluids are also considered.