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Multidimensional stability of planar viscous shock waves. (English) Zbl 0989.35089

Freistühler, Heinrich (ed.) et al., Advances in the theory of shock waves. Boston, MA: Birkhäuser. Prog. Nonlinear Differ. Equ. Appl. 47, 307-516 (2001).
The author presents a comprehensive treatment of shock stability for multidimensional systems of conservation laws, including both multidimensional and regularizing effects, based on the Evans functions techniques. The notable results, the author indicates in the abstract, include: rigorous justification of the inviscid stability criterion of Erpenbeck-Majda; an “inviscid-like” long-wave stability criterion for viscous overcompressive fronts; a refinement distinguishing weak stability/weak instability in the case of surface waves; the extension to relaxation/real viscosity and combustion; sufficient conditions for nonlinear stability of (strictly parabolic) viscous shock fronts; pointwise bounds for scalar shock fronts; and the extension to general systems of the one-dimensional stability index of Gardner-Zumbrun. The author also indicates open problems and several directions for further development in the theory.
For the entire collection see [Zbl 0966.00009].

MSC:

35L67 Shocks and singularities for hyperbolic equations
35L65 Hyperbolic conservation laws
35B35 Stability in context of PDEs
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