or, by (20):
We observe that the coefficient matrices in (23) are Hessian matrices derived from the vector potential
. Therefore the matrices are symmetric.
Also the matrix
is negative definite on account on the concavity (4
) of
with respect to
. This is so, because the defining equation of
, viz.
represents the Legendre transformation from
to
connected with the map
between dual fields. Indeed, we have by (20
,
21
) and (8
)
Such a transformation preserves convexity - or concavity - so
that
is a concave function of
, since
is a concave function of
.
A quasilinear system of the type (23) with symmetric coefficient matrices, of which the temporal one
is definite, is called symmetric hyperbolic. We conclude that
symmetric hyperbolicity of the equations (23
) for the fields
is equivalent to the concavity of the entropy density
in terms of the fields of densities
.
Hyperbolicity implies finite characteristic speeds, and symmetric hyperbolic systems guarantee the well-posedness of initial value problems, i.e. existence and uniqueness of solutions - at least in the neighbourhood of an event - and continuous dependence on the data.
Thus without having actually calculated a single characteristic speed, we have resolved Cattaneo's paradox of infinite speeds. The structure of extended thermodynamics guarantees that all speeds are finite; no paradox can occur!
The fact that a system of balance-type field equations is symmetric hyperbolic, if it is compatible with the entropy inequality and the concavity of the entropy density was discovered by Godunov [19] in the special case of Eulerian fluids. In general this was proved by Boillat [1]. Ruggeri & Strumia [43] have found that the symmetry is revealed only when the Lagrange multipliers are chosen as variables; these authors were strongly motivated by Liu's results of 1972 and by a paper by Friedrichs & Lax [18] which appeared a year earlier.
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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 |