3.2 Symmetric Hyperbolicity3 Finite Speeds in Non-Relativistic 3 Finite Speeds in Non-Relativistic

3.1 Concavity of the Entropy Density 

We recall the argument of Section 2.2 concerning concavity and choose the fields tex2html_wrap_inline3369 to mean the fields of densities tex2html_wrap_inline3353 . Thus equation (8Popup Equation), for A =0, leads to

  equation349

Therefore the concavity of the entropy density tex2html_wrap_inline3415 in the variables tex2html_wrap_inline3353 - the negative-definiteness of tex2html_wrap_inline3653 - implies global invertibility between the field vector tex2html_wrap_inline3353 and the Lagrange multipliers tex2html_wrap_inline3481 .

The transformation tex2html_wrap_inline3613 helps us to recognize the structure of the field equations and to find generic restrictions on the constitutive functions.

Indeed, obviously, with tex2html_wrap_inline3481 as field vector instead of tex2html_wrap_inline3369, or tex2html_wrap_inline3353, we may rephrase (8Popup Equation) in the form

  equation373

where

  equation379

Thus we have

  equation385

and

  equation391

so that the constitutive quantities tex2html_wrap_inline3361 and tex2html_wrap_inline3425 result from tex2html_wrap_inline3687 - defined by equation (19Popup Equation) - through differentiation. Therefore the vector tex2html_wrap_inline3687 is called the thermodynamic vector potential .

It follows from equation (20Popup Equation) that

displaymath404

which implies 4 n (n -1) restrictions on the constitutive functions tex2html_wrap_inline3697 .



3.2 Symmetric Hyperbolicity3 Finite Speeds in Non-Relativistic 3 Finite Speeds in Non-Relativistic

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