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1992 | 27 | 1 | 257-269
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On the problem of symmetrization of hyperbolic equations

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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]).
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  • Institute of Mathematics, Russian Academy of Sciences, Siberian Branch, Universitetskiĭ Prosp. 4, 630090 Novosibirsk, Russia
  • [1] A. M. Blokhin, On symmetrization of the Landau equations in the theory of superfluidity of helium II, Dinamika Sploshn. Sredy 68 (1984), 13-34 (in Russian).
  • [2] A. M. Blokhin, Uniqueness of classical solution of gas dynamics mixed problem with boundary conditions on a shock wave, Sibirsk. Mat. Zh. 23 (5) (1982), 17-30 (in Russian).
  • [3] G. Boillat, Symmétrisation des systèmes d'équations aux dérivées partielles avec densité d'énergie convexe et contraintes, C. R. Acad. Sci. Paris Sér. I 295 (9) (1982), 551-554.
  • [4] M. D. Choi and T. Y. Lam, Extremal positive semidefinite forms, Math. Ann. 231 (1) (1977), 1-18.
  • [5] K. O. Friedrichs, Symmetric hyperbolic linear differential equations, Comm. Pure Appl. Math. 5 (2) (1954), 345-392.
  • [6] L. Gårding, Cauchy's Problem for Hyperbolic Equations, Chicago 1957.
  • [7] S. K. Godunov, Equations of Mathematical Physics, Nauka, Moscow 1979 (in Russian).
  • [8] S. K. Godunov, A symmetric form of the equations of magnetohydrodynamics, Chisl. Metody Mekh. Sploshn. Sredy 3 (1) (1972), 26-34 (in Russian).
  • [9] S. K. Godunov and V. I. Kostin, Transformation of a hyperbolic equation to a symmetric hyperbolic system in the case of two spatial variables, Sibirsk. Mat. Zh. 21 (6) (1980), 3-20.
  • [10] V. M. Gordienko, Un problème mixte pour l'équation vectorielle des ondes: Cas de dissipation de l'énergie; Cas mal posés, C. R. Acad. Sci. Paris Sér. A 288 (10) (1979), 547-550.
  • [11] V. M. Gordienko, Symmetrization of a mixed problem for a second order hyperbolic equation with two spatial variables, Sibirsk. Mat. Zh. 22 (2) (1981), 84-104 (in Russian).
  • [12] D. Hilbert, Über die Darstellung definiter Formen als Summe von Formenquadraten, Math. Ann. 32 (1888), 342-350.
  • [13] L. Hörmander, Linear Partial Differential Operators, Springer, Berlin 1963.
  • [14] V. V. Ivanov, Strictly hyperbolic polynomials which do not admit hyperbolic symmetrization, preprint 77, Inst. of Math., Siberian Branch Acad. Sci. USSR, Novosibirsk 1984.
  • [15] V. I. Kostin, Transformation of a hyperbolic equation to a symmetric system, Ph.D. thesis, Novosibirsk 1981 (in Russian).
  • [16] M. G. Kreĭn and M. A. Naĭmark, Method of symmetric and hermitian forms in the theory of separating roots of algebraic equations, Khar'kov 1936.
  • [17] J. Leray, Lectures on Hyperbolic Equations with Variable Coefficients, Inst. for Adv. Study, Princeton 1952.
  • [18] N. G. Marchuk, On the existence of solutions of a mixed problem for the vector-valued wave equation, Dokl. Akad. Nauk SSSR 252 (3) (1980), 546-550 (in Russian).
  • [19] T. Yu. Mikhailova, Symmetrization of invariant hyperbolic equations, ibid. 270 (3) (1983), 246-250 (in Russian).
  • [20] M. Rosenblatt, A multidimensional prediction problem, Ark. Mat. 3 (5) (1958), 407-424.
  • [21] A. V. Tishchenko, On a basis of solutions of the homogeneous Hörmander identity, Sibirsk. Mat. Zh. 26 (1) (1985), 150-158 (in Russian).
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