×

Energy integrals of hyperbolic equations in the sense of Petrovskij. (Russian. English summary) Zbl 0581.35045

The paper is a survey of results achieved by the Novosibirsk mathematical school in construction of new energy integrals for general strongly hyperbolic equations. One utilizes the nonuniqueness of quadratic forms in Leray’s construction (by a separating operator) to obtain positive definiteness of the first quadratic form. The strongly hyperbolic equation may be reduced to a symmetric hyperbolic system in the sense of Friedrichs if this construction is possible. New energy integrals are used for initial-boundary value problems, e.g. for wave equation and gas dynamics.

MSC:

35L40 First-order hyperbolic systems
35B45 A priori estimates in context of PDEs
35A15 Variational methods applied to PDEs