A note on the non-formation of vacuum states for compressible Navier-Stokes equations

We give a refinement of Lemma 2.2 in [D. Hoff, J.A. Smoller, Non-formation of vacuum states for compressible Navier-Stokes equations, Comm. Math. Phys. 216 (2001) 255-276] and complete the proof of non-formation of vacuum states for one-dimensional compressible Navier-Stokes equation given there.