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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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FUNDAMENTAL IDEAS

FUNDAMENTAL IDEAS

Chapter:
(p.101) 6 FUNDAMENTAL IDEAS
Source:
Dynamics of Viscous Compressible Fluids
Author(s):

Eduard Feireisl

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780198528388.003.0006

Given the rather poor a priori estimates available, the methods of weak convergence play a decisive role in the mathematical theory to be developed in this book. Both ‘classical’ problems of this approach — the presence of oscillations and concentrations in sequences of approximate solutions — are present. The well-known results of the theory of compensated compactness are used in order to cope with possible density oscillations. More specifically, the fundamental properties of the effective viscous pressure discovered by P.-L. Lions are discussed together with an alternative proof of ‘continuity’ of this quantity via the famous div-curl lemma. Next, the concept of oscillation defect measure is introduced, and its relation to the propagation of oscillations and the renormalized continuity equation is established. Furthermore, the whole machinery is applied to the problem of propagation of density oscillations in a sequence of solutions, and it is shown that the oscillations decay in time at a uniform rate independent of the choice of initial data provided the pressure is a monotone function of the density. The weak sequential stability (compactness) of the set of weak solutions is established for optimal values of the ‘adiabatic’ exponent. In particular, the physically interesting case of the monoatomic gas in the isentropic regime in three space dimensions can be treated — a problem left open in current theory. Possible concentrations in the temperature are treated via the method of renormalization (rescaling). A ‘renormalized’ formulation of the thermal energy equation is supplemented with the concept of a renormalized limit, usefulness of which being demonstrated on the problem of weak sequential stability and the study of possible concentrations of the temperature in the thermal energy equation.

Keywords:   effective viscous pressure, oscillations defect measure, biting limit, renormalized limit, propagation of oscillations

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