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IMA Journal of Applied Mathematics Advance Access published online on July 21, 2006
IMA Journal of Applied Mathematics, doi:10.1093/imamat/hxl013
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© The Author 2006. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.
Received September 8, 2004
Accepted March 22, 2006

Article

Self-similar solutions to the isothermal compressible Navier-Stokes equations

Zhenhua Guo 1 * and Song Jiang 2

1 Laboratory of Nonlinear Analysis, Department of Mathematics, Huazhong Normal University, Wuhan 430079, China and LCP, Institute of Applied Physics and Computational Mathematics, PO Box 8009, Beijing 100088, China
2 LCP, Institute of Applied Physics and Computational Mathematics, PO Box 8009, Beijing 100088, China

* To whom correspondence should be addressed.
Zhenhua Guo, E-mail: guo_zhenhua{at}iapcm.ac.cn


  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 Neèas, J., R {udot}zicka, M. & Sverák, 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.

Keywords: Self-similar solutions; Isothermal compressible flows; Navier-Stokes equations.
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