3.3 Moments as Variables3 Finite Speeds in Non-Relativistic 3.1 Concavity of the Entropy

3.2 Symmetric Hyperbolicity 

Using the new variables tex2html_wrap_inline3481 we may write the field equations in the form

  equation414

or, by  (20Popup Equation):

  equation424

We observe that the coefficient matrices in (23Popup Equation) are Hessian matrices derived from the vector potential tex2html_wrap_inline3687 . Therefore the matrices are symmetric.

Also the matrix tex2html_wrap_inline3725 is negative definite on account on the concavity (4Popup Equation) of tex2html_wrap_inline3415 with respect to tex2html_wrap_inline3353 . This is so, because the defining equation of tex2html_wrap_inline3731, viz.

  equation443

represents the Legendre transformation from tex2html_wrap_inline3415 to tex2html_wrap_inline3731 connected with the map tex2html_wrap_inline3613 between dual fields. Indeed, we have by (20Popup Equation, 21Popup Equation) and (8Popup Equation)

  equation455

Such a transformation preserves convexity - or concavity - so that tex2html_wrap_inline3731 is a concave function of tex2html_wrap_inline3481, since tex2html_wrap_inline3415 is a concave function of tex2html_wrap_inline3353 .

A quasilinear system of the type (23Popup Equation) with symmetric coefficient matrices, of which the temporal one is definite, is called symmetric hyperbolic. We conclude that symmetric hyperbolicity of the equations (23Popup Equation) for the fields tex2html_wrap_inline3481 is equivalent to the concavity of the entropy density tex2html_wrap_inline3415 in terms of the fields of densities tex2html_wrap_inline3353 .

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.



3.3 Moments as Variables3 Finite Speeds in Non-Relativistic 3.1 Concavity of the Entropy

image Speeds of Propagation in Classical and Relativistic Extended Thermodynamics
Ingo Müller
http://www.livingreviews.org/lrr-1999-1
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