Symmetric hyperbolicity

3.2 Symmetric hyperbolicity

Using the new variables $Λ$ we may write the field equations in the form

or, by (20

We observe that the coefficient matrices in (23

) 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 (4 ) of $h0$ with respect to $0 F$ . This is so, because the defining equation of $′0 h$ , viz.

represents the Legendre transformation from $h0$ to $h ′0$ connected with the map $F 0 ⇐ ⇒ Λ$ between dual fields. Indeed, we have by (20

, 21

) and (8

)

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 (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 $Λ$ 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.