On the problem of symmetrization of hyperbolic equations
Tom 27 / 1992
Streszczenie
The aspects of symmetrization of hyperbolic equations which will be considered in this review have their own history and are related to some classical results from other areas of mathematics ([12]). Here symmetrization means representation of an initial system of equations in the form of a symmetric t-hyperbolic system in the sense of Friedrichs. Some equations of mathematical physics, for example, the equations of acoustics, of gas dynamics, etc. already have this form. In the 70's S. K. Godunov published a work [8] on a symmetric form of the equations of magnetohydrodynamics. This result was repeated in the 80's ([3]). Later A. M. Blokhin ([1]) got an analogous result for the Landau equations of quantum helium. All the mentioned statements concern systems of equations describing concrete physical objects. One of the motivations for investigating the symmetrization problem comes from the study of initial-boundary value problems for hyperbolic equations. Having a rich set of energy integrals for a given hyperbolic equation one can use them to get estimates of solutions in the well posed problems. Generally one uses a fairly simple theory of initial-boundary value problems with dissipative boundary conditions (see e.g. [7]). This idea has been realized in some simplest cases ([2, 10, 11, 18]).
