Zhikov's hydromechanical lemma on compensated compactness: its extension and application to generalized stationary Navier–Stokes equations

Abstract

In this article, a new variant of the hydromechanical lemma on compensated compactness is proved, the original one is due to Zhikov. As a consequence, new solvability results for the electrorheological system are derived.

Keywords:

• compensated compactness
• div-curl lemma
• generalized Navier–Stokes system
• electrorheological fluids
• coupled system of hydromechanics

AMS Subject Classifications::

• 35Q30
• 35A16
• 35Q35

Acknowledgements

Through this article, dedicated to Prof Zhikov's jubilee, the author expresses her acknowledgement to Prof V.V. Zhikov. The author made the acquaintance of Prof V.V. Zhikov in 2001 but had been already, for a decade or even more, under the influence of his fundamental papers and books, particularly on homogenization. Since 2001, joint researches were started in collaboration with Prof V.V. Zhikov in various fields, homogenization of singular and thin structures, PDEs with degenerate weights and PDEs with variable nonlinearity exponents being among them. This article is in the framework of joint interests. The author was partially supported by RFBR (Russian Foundation of Basic Researches), projects 08-01-00616 and 09-01-12157.