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Planar Stability Criteria for Viscous Shock Waves of Systems with Real Viscosity

Chapter
Part of the Lecture Notes in Mathematics book series (LNM, volume 1911)

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].

Keywords

Shock Wave Resolvent Equation Spectral Stability Branch Singularity Inviscid Case 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  1. 1.Department of MathematicsIndiana UniversityBloomingtonUSA

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