Type of Publication:  Journal article 
Author:  Barker, Blake; Freistühler, Heinrich; Zumbrun, Kevin 
Year of publication:  2015 
Published in:  Archive for Rational Mechanics and Analysis ; 217 (2015), 1.  pp. 309372.  ISSN 00039527.  eISSN 14320673 
DOI (citable link):  https://dx.doi.org/10.1007/s0020501408386 
Summary: 
We consider by a combination of analytical and numerical techniques, some basic questions regarding the relations between inviscid and viscous stability and the existence of a convex entropy. Specifically, for a system possessing a convex entropy, in particular for the equations of gas dynamics with a convex equation of state, we ask: (1) can inviscid instability occur? (2) can viscous instability not detected by inviscid theory occur? (3) can there occur the—necessarily viscous—effect of Hopf bifurcation, or “galloping instability”? and, perhaps most important from a practical point of view, (4) as shock amplitude is increased from the (stable) weakamplitude limit, can there occur a first transition from viscous stability to instability that is not detected by inviscid theory? We show that (1) does occur for strictly hyperbolic, genuinely nonlinear gas dynamics with certain convex equations of state, while (2) and (3) do occur for an artifically constructed system with convex viscositycompatible entropy. We do not know of an example for which (4) occurs, leaving this as a key open question in viscous shock theory, related to the principal eigenvalue property of Sturm Liouville and related operators. In analogy with, and partly proceeding close to, the analysis of Smith on (non)uniqueness of the Riemann problem, we obtain convenient criteria for shock (in)stability in the form of necessary and sufficient conditions on the equation of state.

Subject (DDC):  510 Mathematics 
Keywords:  Mechanics, Physics, general, Theoretical, Mathematical and Computational Physics, Statistical Physics, Dynamical Systems and Complexity, Fluid and Aerodynamics 
Bibliography of Konstanz:  Yes 
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BARKER, Blake, Heinrich FREISTÜHLER, Kevin ZUMBRUN, 2015. Convex Entropy, Hopf Bifurcation, and Viscous and Inviscid Shock Stability. In: Archive for Rational Mechanics and Analysis. 217(1), pp. 309372. ISSN 00039527. eISSN 14320673. Available under: doi: 10.1007/s0020501408386
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