to mean the privileged densities
The privileged entropy density is assumed by (5
) to be concave with respect to the privileged fields
. The privileged co-vector
will be chosen so that the concavity of
implies symmetric hyperbolicity of the field equations.
From (8) we obtain after multiplication by
hence
is still defined as
, as in (19). From (55) it follows that the concavity of
- the negative definiteness of
- implies global invertibility between the field vector
and the Lagrange multipliers
,
provided that
the privileged co-vector
is chosen as co-linear to the vector potential
. We set
Indeed, in that case we have
hence
so that, by (57), the second term on the right hand side of (55) vanishes and
is definite. Equation (58) will be used later.
With
as a field vector, instead of
, we may rephrase (8) in the form
or
hence
where
. Thus
is the Legendre transform of
with respect to the map
. It follows that
is concave in
, since
is concave in
; thus we have
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Speeds of Propagation in Classical and Relativistic
Extended Thermodynamics
Ingo Müller http://www.livingreviews.org/lrr-1999-1 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |
