Self-similar solutions of the compressible flow in one-space dimension. (English) Zbl 1397.35174

Summary: For the isentropic compressible fluids in one-space dimension, we prove that the Navier-Stokes equations with density-dependent viscosity have neither forward nor backward self-similar strong solutions with finite kinetic energy. Moreover, we obtain the same result for the nonisentropic compressible gas flow, that is, for the fluid dynamics of the Navier-Stokes equations coupled with a transport equation of entropy. These results generalize those in Z. Guo and S. Jiang’s work [IMA J. Appl. Math. 71, No. 5, 658–669 (2006; Zbl 1112.76059)] where the one-dimensional compressible fluids with constant viscosity are considered.


35Q30 Navier-Stokes equations
35C06 Self-similar solutions to PDEs
76N99 Compressible fluids and gas dynamics
35D35 Strong solutions to PDEs


Zbl 1112.76059
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