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IMA Journal of Applied Mathematics Advance Access originally published online on July 21, 2006
IMA Journal of Applied Mathematics 2006 71(5):658-669; 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.

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

Zhenhua Guo1,** and Song Jiang2,***

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

** Email: guo_zhenhua{at}iapcm.ac.cn

*** Email: jiang{at}iapcm.ac.cn

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 Necas, 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.

Received on 8 September 2004. accepted on 22 March 2006.


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