Abstract: We consider the Cauchy problem, free boundary problem and piston problem for one-dimensional compressible Navier-Stokes equations with density-dependent viscosity. Using the reduction to absurdity method, we prove that the weak solutions to these systems do not exhibit vacuum states, provided that no vacuum states are present initially. The essential requirements on the solutions are that the mass and energy of the fluid are locally integrable at each time, and the L_loc¹-norm of the velocity gradient is locally integrable in time.

Key words: Compressible Navier-Stokes equations, Vacuum states, Density-dependent viscosity

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