Global helically symmetric solutions to the stokes approximation equations for three-dimensional compressible viscous flows. (English) Zbl 1110.35067

Guo, Zhenhua; Jiang, Song; Li, Jing

Methods Appl. Anal. 12, No. 2, 135-151 (2005).

Summary: 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 \(\mathbb{R}^3\) 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 \(\mathbb{R}^3\) and uniform a priori estimates. The present paper extends 2D existence results of P. L. Lions and Lu, Kazhikhov and Ukai to the three-dimensional helically symmetric case.

MSC:

35Q35	PDEs in connection with fluid mechanics
35Q30	Navier-Stokes equations
76N10	Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics

Keywords:

Navier-Stokes equations; helically symmetric flow; classical solutions; large-time behavior; existence; a priori estimates