Duke Mathematical Journal

Nonclassical multidimensional viscous and inviscid shocks

Olivier Guès, Guy Métivier, Mark Williams, and Kevin Zumbrun

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Abstract

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

Article information

Source
Duke Math. J., Volume 142, Number 1 (2008), 1-110.

Dates
First available in Project Euclid: 27 March 2008

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1206642064

Digital Object Identifier
doi:10.1215/00127094-2008-001

Mathematical Reviews number (MathSciNet)
MR2397883

Zentralblatt MATH identifier
1162.35050

Subjects
Primary: 35L67: Shocks and singularities [See also 58Kxx, 76L05]
Secondary: 76N17: Viscous-inviscid interaction

Citation

Guès, Olivier; Métivier, Guy; Williams, Mark; Zumbrun, Kevin. Nonclassical multidimensional viscous and inviscid shocks. Duke Math. J. 142 (2008), no. 1, 1--110. doi:10.1215/00127094-2008-001. https://projecteuclid.org/euclid.dmj/1206642064


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