Symmetrization of systems of conservation equations and the converse to the condition of Friedrichs and Lax

Abstract

The result of FRIEDRICHS and LAX [Proc. Nat. Acad. Sci. U.S.A., 68, 8, 1686-1688, 1971] concluding that if the system of conservation equations implies the additional conservation equation (balance of entropy) then it can be symmetrized by premultiplication by the Hessian matrix of the entropy, is well known. Basic ingredients of the proof of the converse to this result can be found in the paper by BOILLAT [C.R. Acad. Sci. Paris, 278 A, 909-914 1974], however this converse has not been explicitly formulated there and, as a consequence, it seems to be overlooked. Therefore, an explicit formulation and the detailed proof of the converse to the condition of Friedrichs and Lax is given in this paper. Due to this result, the restrictions imposed on the system of conservation equations by consistency with the additional conservation equations can be alternatively derived from requirement that it admits Hessian matrix as a symmetrizer while the corresponding entropies can be determined by direct integration of the admissible Hessian symmetrizers. As an illustrative example, the system of conservation equations given in [DOMAŃSKI, JABŁOŃSKI and KOSIŃSKI, Arch. Mech., 48, 541-550, 1996] is analysed. It is shown that this system can be brought into equivalent symmetric hyperbolic form without appealing to the existence of the additional conservation equation and the whole family of symmetric symmetrizers is determined. Then, the condition that the system admits the additional conservation equation reduces to the requirement that the family of symmetric symmetrizers contains at least one Hessian matrix. This requirement is, in turn, equivalent to the integrability condition for the overdetermined system of second order partial differential equations for the scalar entropy function. Finally, the family of entropies is obtained as a result of integration of this system.