Blowup of Smooth Solutions to the Navier-Stokes Equations for   Compressible Isothermal Fluids
  It is shown that the one-dimensional or two-dimensional radially symmetric isothermal compressible Navier-Stokes system has no non-trivial global smooth solutions if the initial density is compactly supported. This result is a generalization of Xin's work \cite{Xin98} to the isothermal case. 
Du Dapeng, Li Jingyu and Zhang Kaijun
math.AP