15 March 2008 Nonclassical multidimensional viscous and inviscid shocks
Olivier Guès, Guy Métivier, Mark Williams, Kevin Zumbrun
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Duke Math. J. 142(1): 1-110 (15 March 2008). DOI: 10.1215/00127094-2008-001


Extending our earlier work on Lax-type shocks of systems of conservation laws (see [GM+1], [GM+2], [GM+4]), we establish existence and stability of curved multidimensional shock fronts in the vanishing viscosity limit for general Lax- or undercompressive-type shock waves of nonconservative hyperbolic systems with parabolic regularization. The hyperbolic equations may be of variable multiplicity, and the parabolic regularization may be of “real,” or partially parabolic, type. We prove an existence result for inviscid nonconservative shocks which extends a one-dimensional result of X to multidimensional shocks. Lin [L] proved by quite different methods. In addition, we construct families of smooth viscous shocks converging to a given inviscid shock as viscosity goes to zero, thereby justifying the small viscosity limit for multidimensional nonconservative shocks.

In our previous work on shocks, we made use of conservative form, especially in parts of the low-frequency analysis. Thus, most of the new analysis of this article is concentrated in this area. By adopting the more general nonconservative viewpoint, we are able to shed new light on both the viscous and inviscid theories. For example, we can now provide a clearer geometric motivation for the low-frequency analysis in the viscous case. Also, we show that one may, in the treatment of inviscid stability of nonclassical and/or nonconservative shocks, remove an apparently restrictive technical assumption made by Mokrane [Mo] and Coulombel [C] in their work on, respectively, shock-type nonconservative boundary problems and conservative undercompressive shocks. Another advantage of the nonconservative perspective is that Lax and undercompressive shocks can be treated by exactly the same analysis.


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Olivier Guès. Guy Métivier. Mark Williams. Kevin Zumbrun. "Nonclassical multidimensional viscous and inviscid shocks." Duke Math. J. 142 (1) 1 - 110, 15 March 2008. https://doi.org/10.1215/00127094-2008-001


Published: 15 March 2008
First available in Project Euclid: 27 March 2008

zbMATH: 1162.35050
MathSciNet: MR2397883
Digital Object Identifier: 10.1215/00127094-2008-001

Primary: 35L67
Secondary: 76N17

Rights: Copyright © 2008 Duke University Press


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Vol.142 • No. 1 • 15 March 2008
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