Abstract
We investigate the self-similar solutions to the isothermal compressible Navier-Stokes equations. The aim of this paper is to show that there exist neither forward nor backward self-similar solutions with finite total energy. This generalizes the results for the incompressible case in Necas, J., Ruzicka, M. & Sverak, V. (1996, On Leray's self-similar solutions of the Navier-Stokes equations. Acta. Math., 176, 283-294), and is consistent with the (unproved) existence of regular solutions globally in time for the compressible Navier-Stokes equations.
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