3.2 Symmetric hyperbolicity

Using the new variables Λ we may write the field equations in the form
∂F A -----Λ,A = π(Λ ), (22 ) ∂Λ
or, by  (20View Equation):
∂2h ′A ------Λ,A = π(Λ ). (23 ) ∂Λ ∂Λ
We observe that the coefficient matrices in (23View Equation) are Hessian matrices derived from the vector potential h ′A. Therefore the matrices are symmetric.

Also the matrix ∂2h′0 ------- ∂Λ ∂Λ is negative definite on account on the concavity (4View Equation) of h0 with respect to 0 F. This is so, because the defining equation of ′0 h, viz.

′0 0 0 h = Λ ⋅ F − h (24 )
represents the Legendre transformation from h0 to h ′0 connected with the map F 0 ⇐ ⇒ Λ between dual fields. Indeed, we have by (20View Equation, 21View Equation) and (8View Equation)
0 ∂h-′0 -∂h0- F = ∂ Λ and Λ = ∂F 0 . (25 )
Such a transformation preserves convexity – or concavity – so that ′0 h is a concave function of Λ, since h0 is a concave function of F 0.

A quasilinear system of the type (23View Equation) with symmetric coefficient matrices, of which the temporal one is definite, is called symmetric hyperbolic. We conclude that symmetric hyperbolicity of the equations (23View Equation) for the fields Λ is equivalent to the concavity of the entropy density 0 h in terms of the fields of densities 0 F.

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.

  Go to previous page Go up Go to next page