Methods and Applications of Analysis

Volume 12 (2005)

Number 2

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

Pages: 135 – 152



Zhenhua Gao

Song Jiang

Jing Li


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) $\gamma > 1$ 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.


Stokes approximation equations; helically symmetric flow; classical solutions

2010 Mathematics Subject Classification

35Q30, 35Q35

Published 1 January 2005