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A study of strong discontinuities stability in continuum mechanics. (English) Zbl 0898.76038

This article is a survey of several papers by the author and his collaborators, dealing with the multi-dimensional stability of strong discontinuities such as shock waves. The outlined methodology is as follows. First, a symmetrization process is applied to systems of conservation laws that may be associated with divergence-free conditions, such as the MHD equations, for instance. This process is a generalization of Godunov’s formalism. Then, through an approach similar to Majda’s [A. Majda, Compressible fluid flow and systems of conservation laws in several space variables. Applied Mathematical Sciences, 53. New York etc.: Springer-Verlag VIII, 159 p. (1984; Zbl 0537.76001)], a linearization of the quasilinear equations as well as the jump conditions lead to a linear initial-boundary value problem, for which a Lopatinsky condition is derived. This condition does not ensure in general that the boundary conditions are dissipative. However, the author shows that an extended system is strictly dissipative. This enables him, by means of the dissipative integral technique, to derive a priori estimates. These estimates involve a loss of derivatives in the case when the Lopatinsky condition is satisfied but not the uniform condition. Several interesting applications are mentioned, concerning classical or relativistic gas dynamics, MHD, superfluidity, anisotropic plasmas.
Reviewer: S.Benzoni (Lyon)

MSC:

76E99 Hydrodynamic stability
76L05 Shock waves and blast waves in fluid mechanics
76-02 Research exposition (monographs, survey articles) pertaining to fluid mechanics

Citations:

Zbl 0537.76001
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