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The stability of the solutions of some boundary-value problems for hyperbolic equations. (English. Russian original) Zbl 0812.35075
J. Appl. Math. Mech. 56, No. 1, 36-45 (1992); translation from Prikl. Mat. Mekh. 56, No. 1, 40-51 (1992).
The authors investigate the behaviour of the solutions of linear hyperbolic equations as \(t\to\infty\) in the half space \(x>0\), \(y_ i\in\mathbb{R}\), \(i=1,2,\dots, r\), with boundary conditions given for \(x=0\). The equation and boundary conditions are assumed to be homogeneous with respect to the order of differentiation and all coefficients are assumed to be constant. It is shown, that the disturbances may have the following type of behaviour as \(t\to\infty\): exponentially increasing (instability), decay as a power function (stability) and remains bounded (neutral stability). The transitions of the system to an unstable, stable and neutrally stable state are investigated and the criteria for these transitions are given. These criteria are used to establish the existence of neutrally stable magnetohydrodynamic shocks even in the case of an ideal gas, a phenomenon that has not been previously documented. The existence of an a priori bound on the solution has been proved for these systems in cases of stability and neutral stability. The interaction of disturbances with the boundary in the case of neutral stability produces a non-smooth solution, so that the a priori bound is unimprovable. The behaviour of disturbances on the boundary is described by the solution of the Cauchy problem for some systems of partial differential equations of a high order with special conditions on the external forces and the initial values.
Reviewer: A.Doktor (Praha)
MSC:
35L50 Initial-boundary value problems for first-order hyperbolic systems
35B35 Stability in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
76L05 Shock waves and blast waves in fluid mechanics
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References:
[1] D’yakov, S. P.: On the stability of shock waves. Zh. eksp. Teor. fiz. 27, No. 3, 288-295 (1954)
[2] Iordanskii, S. V.: On the stability of a plane stationary shock wave. Prikl. mat. Mekh. 21], No. 4, 465-472 (1957)
[3] Kantorovich, V. M.: On the stability of shock waves. Zh. eksp. Tear. fiz. 33, No. 6, 1525-1526 (1957)
[4] Erpenbeck, J. J.: Stability of step shocks. Phys. fluids 5, No. 10, 1181-1187 (1962) · Zbl 0111.38403
[5] Gardner, C. S.; Kruskal, M. D.: Stability of plane magnetohydrodynamic shocks. Phys. fluids 7, No. 5, 700-706 (1964) · Zbl 0121.21107
[6] Filippova, O. L.: Stability of plane MHD shocks. Proceedings of the 6th all-union conf. On theoretical and applied mechanics, 616 (1986) · Zbl 0616.73090
[7] Blokhin, A. M.; Druzhinin, I. Yu.: Some well-posed linear problems of the stability of strong shocks in magnetohydrodynamics. Sib. mat. Zh. 31, No. 2, 3-8 (1990) · Zbl 0712.76096
[8] Kreiss, H. -O.: Initial boundary value problems for hyperbolic systems. Comm. pure appl. Math. 23, No. 3, 277-296 (1970) · Zbl 0188.41102
[9] Majda, A.; Osher, S.: Initial-boundary value problems for hyperbolic equations with uniformly characteristic boundary. Comm. pure appl. Math. 28, No. 5, 607-675 (1975) · Zbl 0314.35061
[10] Mikhailov, Yu.Ya.; Chinilov, A. Yu.: Construction of a solution of a linear stability problem for a shock wave. Mzhg4, 130-138 (1988)
[11] Sobolev, S. L.: On mixed problems for partial differential equations with two unknown variables. Dokl. akad. Nauk SSSR 122, No. 4, 55-558 (1958) · Zbl 0113.08301
[12] Kurosh, A. G.: A course in higher algebra. (1975) · Zbl 0038.15104
[13] Kulikovskii, A. G.; Lyubimov, G. A.: Magnetohydrodynamics. (1962) · Zbl 0116.42906
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