Duke Mathematical Journal

Nonclassical multidimensional viscous and inviscid shocks

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

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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

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

First available in Project Euclid: 27 March 2008

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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


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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  • A. V. Azevedo, D. Marchesin, B. J. Plohr, and K. Zumbrun, Nonuniqueness of solutions of Riemann problems, Z. Angew. Math. Phys. 47 (1996), 977--998.
  • —, Bifurcation of nonclassical viscous shock profiles from the constant state, Comm. Math. Phys. 202 (1999), 267--290.
  • S. Benzoni-Gavage, Stability of multi-dimensional phase transitions in a van der Waals fluid, Nonlinear Anal. 31 (1998), 243--263.
  • S. Bianchini, On the Riemann problem for non-conservative hyperbolic systems, Arch. Ration. Mech. Anal. 166 (2003), 1--26.
  • S. Bianchini and A. Bressan, Vanishing viscosity solutions of nonlinear hyperbolic systems, Ann. of Math. (2) 161 (2005), 223--342.
  • L. Q. Brin, Numerical testing of the stability of viscous shock waves, Math. Comp. 70 (2001), 1071--1088.
  • L. Q. Brin and K. Zumbrun, ``Analytically varying eigenvectors and the stability of viscous shock waves'' in Seventh Workshop on Partial Differential Equations, Part I (Rio de Janeiro, 2001), Mat. Contemp. 22, Soc. Brasil. Mat., Rio de Janeiro, 2002, 19, --32.
  • J. Chazarain and A. Piriou, Introduction to the Theory of Linear Partial Differential Equations, Stud. Math. Appl. 14, North-Holland, Amsterdam, 1982.
  • J.-F. Coulombel, Stability of multidimensional undercompressive shock waves, Interfaces Free Bound. 5 (2003), 367--390.
  • H. FreistüHler, Dynamical stability and vanishing viscosity: A case study of a non-strictly hyperbolic system, Comm. Pure Appl. Math. 45 (1992), 561--582.
  • —, A short note on the persistence of ideal shock waves, Arch. Math. (Basel) 64 (1995), 344--352.
  • H. FreistüHler and T.-P. Liu, Nonlinear stability of overcompressive shock waves in a rotationally invariant system of viscous conservation laws, Comm. Math. Phys. 153 (1993), 147--158.
  • H. FreistüHler and P. Szmolyan, Spectral stability of small shock waves, Arch. Ration. Mech. Anal. 164 (2002), 287--309.
  • O. GuèS, G. MéTivier, M. Williams, and K. Zumbrun, Existence and stability of multidimensional shock fronts in the vanishing viscosity limit, Arch. Ration. Mech. Anal. 175 (2004), 151--244.
  • —, Multidimensional viscous shocks, II: The small viscosity problem, Comm. Pure Appl. Math. 57 (2004), 141--218.
  • —, Multidimensional viscous shocks, I: Degenerate symmetrizers and long time stability, J. Amer. Math. Soc. 18 (2005), 61--120.
  • —, Navier-Stokes regularization of multidimensional Euler shocks, Ann. Sci. École Norm. Sup. (4) 39 (2006), 75, --175.
  • —, Viscous boundary value problems for symmetric systems with variable multiplicities, J. Differential Equations 244 (2008), 309, --387.
  • —, Paradifferential boundary conditions and persistence of nonclassical shock waves, in preparation.
  • O. GuèS and M. Williams, Curved shocks as viscous limits: A boundary problem approach, Indiana Univ. Math. J. 51 (2002), 421--450.
  • J. Humpherys and K. Zumbrun, An efficient shooting algorithm for Evans function calculations in large systems, Phys. D 220 (2006), 116--126.
  • E. Isaacson, D. Marchesin, and B. J. Plohr, Transitional waves for conservation laws, SIAM J. Math. Anal. 21 (1990), 837--866.
  • T. T. Li and W. Yu, Boundary Value Problems for Quasilinear Hyperbolic Systems, Duke Univ. Math. Ser. 5, Duke Univ., Durham, N.C., 1985.
  • X.-B. Lin, ``Generalized Rankine-Hugoniot condition and shock solutions for quasilinear hyperbolic systems'' in Special Issue in Celebration of Jack K. Hale's 70th Birthday (Atlanta and Lisbon, 1998), J. Differential Equations 168, Elsevier, San Diego, Calif., 2000, 321--354.
  • A. Majda, The existence of multidimensional shock fronts, Mem. Amer. Math. Soc. 43 (1983), no. 281.
  • —, The stability of multidimensional shock fronts, Mem. Amer. Math. Soc. 41 (1983), no. 275.
  • G. MéTivier, Small Viscosity and Boundary Layer Methods, Model. Simul. Sci. Eng. Technol., Birkhäuser, Boston, 2004.
  • G. MéTivier and K. Zumbrun, Symmetrizers and continuity of stable subspaces for parabolic-hyperbolic boundary value problems, Discrete Contin. Dyn. Syst. 11 (2004), 205--220.
  • —, Hyperbolic boundary value problems for symmetric systems with variable multiplicities, J. Differential Equations 211 (2005), 61--134.
  • —, Large viscous boundary layers for noncharacteristic nonlinear hyperbolic problems, Mem. Amer. Math. Soc. 175 (2005), no. 826.
  • A. Mokrane, Problèmes mixtes hyperboliques non linéaires, Ph.D. dissertation, Université de Rennes 1, Rennes, France, 1987.
  • R. Plaza and K. Zumbrun, An Evans function approach to spectral stability of small-amplitude shock profiles, Discrete Contin. Dyn. Syst. 10 (2004), 885--924.
  • P.-A. Raviart and L. Sainsaulieu, A nonconservative hyperbolic system modeling spray dynamics, I: Solution of the Riemann problem, Math. Models Methods Appl. Sci. 5 (1995), 297--333.
  • L. Sainsaulieu, Traveling-wave solutions of convection-diffusion systems in nonconservation form, SIAM J. Math. Anal. 27 (1996), 1286--1310.
  • R. Saurel, S. Gavrilyuk, and F. Renaud, A multiphase model with internal degrees of freedom: Application to shock-bubble interaction, J. Fluid Mech. 495 (2003), 283--321.
  • S. Schecter, Traveling-wave solutions of convection-diffusion systems by center manifold reduction, Nonlinear Anal. 49 (2002), 35--59.
  • S. Schecter and M. Shearer, ``Transversality for undercompressive shocks in Riemann problems'' in Viscous Profiles and Numerical Methods for Shock Waves (Raleigh, N.C., 1990), SIAM, Philadelphia, 1991, 142--154.
  • N. Seguin, Modélisation et simulation numérique des écoulements diphasiques, Ph.D. dissertation, Université de Provence Aix Marseille 1, Marseille, 2002.
  • M. Shearer, The Riemann problem for $2\times 2$ systems of hyperbolic conservation laws with case I quadratic nonlinearities, J. Differential Equations 80 (1989), 343--363.
  • M. Shearer and S. Schecter, ``Undercompressive shocks in systems of conservation laws'' in Nonlinear Evolution Equations That Change Type, IMA Vol. Math. Appl. 27, Springer, New York, 1990, 218--231.
  • M. Slemrod, Admissibility criteria for propagating phase boundaries in a van der Waals fluid, Arch. Rational Mech. Anal. 81 (1983), 301--315.
  • H. B. Stewart and B. Wendroff, Two-phase flow: Models and methods, J. Comput. Phys. 56 (1984), 363--409.
  • J. A. Trangenstein and P. Colella, A higher-order Godunov method for modeling finite deformation in elastic-plastic solids, Comm. Pure Appl. Math. 44 (1991), 41--100.
  • K. Zumbrun, ``Multidimensional stability of planar viscous shock waves'' in Advances in the Theory of Shock Waves, Progr. Nonlinear Differential Equations Appl. 47, Birkhäuser, Boston, 2001, 307--516.
  • K. Zumbrun and D. Serre, Viscous and inviscid stability of multidimensional planar shock fronts, Indiana Univ. Math. J. 48 (1999), 937--992.