Summary
We present a streamlined account of recent developments in the stability theory for planar viscous shock waves, with an emphasis on applications to physical models with “real,” or partial viscosity. The main result is the establishment of necessary, or “weak”, and sufficient, or “strong”, conditions for nonlinear stability analogous to those established by Majda [M.1, M.2, M.3] in the inviscid case but (generically) separated by a codimension-one set in parameter space rather than an open set as in the inviscid case. The importance of codimension one is that transition between nonlinear stability and instability is thereby determined, lying on the boundary set between the open regions of strong stability and strong instability (the latter defined as failure of weak stability). Strong stability holds always for small-amplitude shocks of classical “Lax” type [PZ, FreS]; for large-amplitude shocks, however, strong instability may occur [ZS, Z.3].
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© 2007 Springer-Verlag Berlin Heidelberg
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Zumbrun, K. (2007). Planar Stability Criteria for Viscous Shock Waves of Systems with Real Viscosity. In: Marcati, P. (eds) Hyperbolic Systems of Balance Laws. Lecture Notes in Mathematics, vol 1911. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-72187-1_4
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DOI: https://doi.org/10.1007/978-3-540-72187-1_4
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-72186-4
Online ISBN: 978-3-540-72187-1
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