Global Helically Symmetric Solutions for the Stokes Approximation Equations for Three-Dimensional Compressible Viscous Flows

We prove the existence and uniqueness of global strong solutions to the Cauchy problem of the compressible Stokes approximation equations for any (specific heat ratio) $\gamma \gt 1$ in $\Bbb R^3$ when initial data are helically symmetric. Moreover, the large-time behavior of the strong solution and the existence of global weak solutions are obtained simultaneously. The proof is based on a Ladyzhenskaya interpolation type inequality for helically symmetric functions in $\Bbb R^3$ and uniform a priori estimtes. The present paper extends Lions' and Lu, Kazhikhov and Ukai's existence theorem in $\Bbb R^2$ to the three-dimensional helically symmetric case.

Keywords

Stokes approximation equations, helically symmetric flow, classical solutions

2010 Mathematics Subject Classification

35Q30, 35Q35

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Published 1 January 2005