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Dynamics of Viscous Compressible Fluids$
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Eduard Feireisl

Print publication date: 2003

Print ISBN-13: 9780198528388

Published to Oxford Scholarship Online: September 2007

DOI: 10.1093/acprof:oso/9780198528388.001.0001

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MATHEMATICAL PRELIMINARIES

MATHEMATICAL PRELIMINARIES

Chapter:
(p.20) 2 MATHEMATICAL PRELIMINARIES
Source:
Dynamics of Viscous Compressible Fluids
Author(s):

Eduard Feireisl

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780198528388.003.0002

The basic mathematical concepts used in the book are presumed in this chapter. The necessary function spaces framework — Lebesgue spaces, Sobolev spaces, and spaces of continuous functions — are introduced here. In order to underline the physical background of the function spaces, the exact definitions introduced in Chapter 2 are followed by the energy estimates deduced in Chapter 3 directly for any smooth solution of the underlying equations (a priori estimates). Several other fundamental concepts are also treated at length: average continuity of weak solutions, renormalized solutions of the continuity equation, and instantaneous values of the state variables.

Keywords:   Sobolev space, vector function, weak convergence, weak continuity

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