4.2 Symmetric Hyperbolicity4 Finite Speeds in Relativistic 4 Finite Speeds in Relativistic

4.1 Concavity of a Privileged Entropy Density 

We recall the arguments of Section  2.2 concerning concavity in the relativistic case and choose the fields tex2html_wrap_inline3369 to mean the privileged densities tex2html_wrap_inline3993 The privileged entropy density is assumed by (5Popup Equation) to be concave with respect to the privileged fields tex2html_wrap_inline3995 . The privileged co-vector tex2html_wrap_inline3469 will be chosen so that the concavity of tex2html_wrap_inline3999 implies symmetric hyperbolicity of the field equations.

From (8Popup Equation) we obtain after multiplication by tex2html_wrap_inline3469

  equation814

hence

  equation825

tex2html_wrap_inline3687 is still defined as tex2html_wrap_inline4023, as in (19Popup Equation). From (55Popup Equation) it follows that the concavity of tex2html_wrap_inline4025 - the negative definiteness of tex2html_wrap_inline4027 - implies global invertibility between the field vector tex2html_wrap_inline3995 and the Lagrange multipliers tex2html_wrap_inline3481, provided that the privileged co-vector tex2html_wrap_inline4033 is chosen as co-linear to the vector potential tex2html_wrap_inline3687 . We set

  equation862

Indeed, in that case we have

  equation869

hence

  equation876

so that, by (57Popup Equation), the second term on the right hand side of (55Popup Equation) vanishes and tex2html_wrap_inline4047 is definite. Equation (58Popup Equation) will be used later.

With tex2html_wrap_inline3481 as a field vector, instead of tex2html_wrap_inline3995, we may rephrase (8Popup Equation) in the form

  equation902

or

  equation908

hence

  equation914

where tex2html_wrap_inline4065 . Thus tex2html_wrap_inline4067 is the Legendre transform of tex2html_wrap_inline3999 with respect to the map tex2html_wrap_inline4071 . It follows that tex2html_wrap_inline4067 is concave in tex2html_wrap_inline3481, since tex2html_wrap_inline3999 is concave in tex2html_wrap_inline3995 ; thus we have

  equation938



4.2 Symmetric Hyperbolicity4 Finite Speeds in Relativistic 4 Finite Speeds in Relativistic

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