×

On stability of shock waves in magnetohydrodynamics with anisotropic pressure. (English. Russian original) Zbl 0808.76098

Sib. Math. J. 34, No. 6, 1005-1016 (1993); translation from Sib. Mat. Zh. 34, No. 6, 10-22 (1993).
The article is devoted to the study of well-posedness for some linear mixed stability problems for shock waves in magnetohydrodynamics with anisotropic pressure. We derive an a priori estimate for a solution to the stability problem for a fast parallel shock wave in anisotropic magnetohydrodynamics with high pressure. A similar result is discussed for the stability problem for a fast transverse shock wave. Finally, we study instability of a slow magnetohydrodynamic shock wave in magnetohydrodynamics with anisotropic pressure; namely, we construct an example of ill-posedness similar to Hadamard’s example for the stability problems for a slow parallel shock wave in the case of a cold plasma.

MSC:

76W05 Magnetohydrodynamics and electrohydrodynamics
76L05 Shock waves and blast waves in fluid mechanics
35R25 Ill-posed problems for PDEs
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] A. M. Blokhin and D. A. Krymskikh, ?Statement of stability problems for shock waves in magnetohydrodynamics with anisotropic pressure,? in: Some Applications of Functional Analysis to Equations of Mathematical Physics [in Russian], Inst. Mat. (Novosibirsk), Novosibirsk, 1990, pp. 3-30.
[2] A. M. Blokhin and I. Yu. Druzhinin, ?On the stability of a fast magnetohydrodynamical shock wave for a weak field,? in: Partial Differential Equations [in Russian], Inst. Mat. (Novosibirsk), Novosibirsk, 1989, pp. 15-32.
[3] A. M. Blokhin, Energy Integrals and Their Applications to the Problems of Gas Dynamics [in Russian], Nauka, Novosibirsk (1986).
[4] A. M. Blokhin and I. Yu. Druzhinin, ?On the stability of shock waves in magnetohydrodynamics,? Sibirsk. Mat. Zh.,30, No. 4, 13-29 (1989).
[5] S. Mizohata, The Theory of Partial Differential Equations [Russian translation], Mir, Moscow (1977).
[6] R. Bellman, Introduction to Matrix Analysis [Russian translation], Nauka, Moscow (1976).
[7] V. P. Mikhaîlov, Partial Differential Equations [in Russian], Nauka, Moscow (1976).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.