Navier-Stokes regularization of multidimensional Euler shocks
Guès, C. M. I. Olivier ; Métivier, Guy ; Williams, Mark ; Zumbrun, Kevin
Annales scientifiques de l'École Normale Supérieure, Tome 39 (2006), p. 75-175 / Harvested from Numdam
@article{ASENS_2006_4_39_1_75_0,
     author = {Gu\`es, C. M. I. Olivier and M\'etivier, Guy and Williams, Mark and Zumbrun, Kevin},
     title = {Navier-Stokes regularization of multidimensional Euler shocks},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     volume = {39},
     year = {2006},
     pages = {75-175},
     doi = {10.1016/j.ansens.2005.12.002},
     zbl = {05037727},
     language = {en},
     url = {http://dml.mathdoc.fr/item/ASENS_2006_4_39_1_75_0}
}
Guès, C. M. I. Olivier; Métivier, Guy; Williams, Mark; Zumbrun, Kevin. Navier-Stokes regularization of multidimensional Euler shocks. Annales scientifiques de l'École Normale Supérieure, Tome 39 (2006) pp. 75-175. doi : 10.1016/j.ansens.2005.12.002. http://gdmltest.u-ga.fr/item/ASENS_2006_4_39_1_75_0/

[1] Alinhac S., Existence d'ondes de raréfaction pour des sytèmes quasi-linéaires hyperboliques multidimensionnels, Comm. Partial Differential Equations 14 (1989) 173-230. | MR 976971 | Zbl 0692.35063

[2] Bony J.M., Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires, Ann. Sci. École Norm. Sup. Paris 14 (1981) 209-246. | Numdam | MR 631751 | Zbl 0495.35024

[3] Coppel W.A., Stability and Asymptotic Behavior of Differential Equations, D.C. Heath, Boston, 1965. | MR 190463 | Zbl 0154.09301

[4] Chazarain J., Piriou A., Introduction to the Theory of Linear Partial Differential Equations, North-Holland, Amsterdam, 1982. | MR 678605 | Zbl 0487.35002

[5] Erpenbeck J.J., Stability of step shocks, Phys. Fluids 5 (1962) 1181-1187. | MR 155515 | Zbl 0111.38403

[6] Freistühler H., Szmolyan P., Spectral stability of small shock waves, Arch. Rat. Mech. Anal. 164 (2002) 287-309. | MR 1933630 | Zbl 1018.35010

[7] Gardner R., Zumbrun K., The gap lemma and geometric criteria instability of viscous shock profiles, CPAM 51 (1998) 797-855. | MR 1617251 | Zbl 0933.35136

[8] Gilbarg D., The existence and limit behavior of the one-dimensional shock layer, Amer. J. Math. 73 (1951) 256-274. | MR 44315 | Zbl 0044.21504

[9] Guès O., Développements asymptotiques de solutions exactes de systèmes hyperboliques quasilinéaires, Asymp. Anal. 6 (1993) 241-270. | MR 1201195 | Zbl 0780.35017

[10] Guès O., Métivier G., Williams M., Zumbrun K., Multidimensional viscous shocks I: degenerate symmetrizers and long time stability, J. Amer. Math. Soc. 18 (2005) 61-120. | MR 2114817 | Zbl 1058.35163

[11] Guès O., Métivier G., Williams M., Zumbrun K., Multidimensional viscous shocks II: the small viscosity problem, Comm. Pure Appl. Math. 57 (2004) 141-218. | MR 2012648 | Zbl 1073.35162

[12] Guès O., Métivier G., Williams M., Zumbrun K., Existence and stability of multidimensional shock fronts in the vanishing viscosity limit, Arch. Rat. Mech. Anal. 175 (2004) 151-244. | MR 2118476 | Zbl 1072.35122

[13] Guès O., Métivier G., Williams M., Zumbrun K., Stability of noncharacteristic boundary layers for the compressible Navier-Stokes and MHD equations, in preparation.

[14] Guès O., Métivier G., Williams M., Zumbrun K., Viscous boundary value problems for symmetric systems with variable multiplicities, in preparation.

[15] Goodman J., Nonlinear asymptotic stability of viscous shock profiles for conservation laws, Arch. Rat. Mech. Anal. 95 (1986) 325-344. | MR 853782 | Zbl 0631.35058

[16] Goodman J., Xin Z., Viscous limits for piecewise smooth solutions to systems of conservation laws, Arch. Rat. Mech. Anal. 121 (1992) 235-265. | MR 1188982 | Zbl 0792.35115

[17] Guès O., Williams M., Curved shocks as viscous limits: a boundary problem approach, Indiana Univ. Math. J. 51 (2002) 421-450. | MR 1909296 | Zbl 1046.35072

[18] Hörmander L., The Analysis of Linear Partial Differential Operators I, Springer, Berlin, 1983. | Zbl 1028.35001

[19] Kreiss H.-O., Initial boundary value problems for hyperbolic systems, Comm. Pure Appl. Math. 23 (1970) 277-298. | MR 437941 | Zbl 0193.06902

[20] Kato T., Perturbation Theory for Linear Operators, Springer, Berlin, 1985. | MR 1335452 | Zbl 0148.12601

[21] Kawashima S., Systems of hyperbolic-parabolic type with applications to the equations of magnetohydrodynamics, PhD thesis, Kyoto University, 1983.

[22] Kawashima S., Shizuta Y., Systems of equations of hyperbolic-parabolic type with applications to the discrete Boltzmann equation, Hokkaido Math. J. 14 (1985) 249-275. | MR 798756 | Zbl 0587.35046

[23] Kawashima S., Shizuta Y., On the normal form of the symmetric hyperbolic-parabolic systems associated with the conservation laws, Tohoku Math. J. (1988) 449-464. | MR 957056 | Zbl 0699.35171

[24] Mascia C., Zumbrun K., Pointwise Green function bounds for shock profiles with degenerate viscosity, Arch. Rat. Mech. Anal. 169 (2003) 177-263. | MR 2004135 | Zbl 1035.35074

[25] Majda A., Pego R., Stable viscosity matrices for systems of conservation laws, J. Differential Equations 56 (1985) 229-262. | MR 774165 | Zbl 0512.76067

[26] Majda A., The Stability of Multidimensional Shock Fronts, Mem. Amer. Math. Soc., vol. 275, AMS, Providence, RI, 1983. | MR 683422 | Zbl 0506.76075

[27] Majda A., The Existence of Multidimensional Shock Fronts, Mem. Amer. Math. Soc., vol. 281, AMS, Providence, RI, 1983. | MR 699241 | Zbl 0517.76068

[28] Métivier G., Stability of Multidimensional Shocks, in: Advances in the Theory of Shock Waves, Progress in Nonlinear PDE, vol. 47, Birkhäuser, Boston, 2001. | MR 1842775 | Zbl 1017.35075

[29] Métivier G., The block structure condition for symmetric hyperbolic systems, Bull. Lond. Math. Soc. 32 (2000) 689-702. | MR 1781581 | Zbl 1073.35525

[30] Métivier G., Zumbrun K., Large viscous boundary layers for noncharacteristic nonlinear hyperbolic problems, Mem. Amer. Math. Soc. 175 (826) (2005), vi+107 p. | MR 2130346 | Zbl 1074.35066

[31] Métivier G., Zumbrun K., Symmetrizers and continuity of stable subspaces for parabolic-hyperbolic boundary value problems, Disc. Cont. Dyn. Syst. 11 (2004) 205-220. | MR 2073953 | Zbl 1102.35332

[32] Métivier G., Zumbrun K., Hyperbolic boundary value problems for symmetric systems with variable multiplicities, J. Differential Equations 211 (2005) 61-134. | MR 2121110 | Zbl 1073.35155

[33] Pego R., Stable viscosities and shock profiles for systems of conservation laws, Trans. Amer. Math. Soc. 282 (1984) 749-763. | MR 732117 | Zbl 0512.76068

[34] Plaza R., Zumbrun K., An Evans function approach to spectral stability of small-amplitude shock profiles, J. Disc. Cont. Dyn. Syst. 10 (2004) 885-924. | MR 2073940 | Zbl 1058.35164

[35] Steenrod N., The Topology of Fibre Bundles, Princeton University Press, Princeton, NJ, 1951. | MR 39258 | Zbl 0054.07103

[36] Zumbrun K., Multidimensional stability of planar viscous shock waves, in: Advances in the Theory of Shock Waves, Progress in Nonlinear PDE, vol. 47, Birkhäuser, Boston, 2001, pp. 304-516. | MR 1842778 | Zbl 0989.35089

[37] Zumbrun K., Stability of large-amplitude shock waves of compressible Navier-Stokes equations, with an appendix by H.K. Jenssen and G. Lyng, in: Handbook of Mathematical Fluid Dynamics, vol. 3, North-Holland, Amsterdam, 2004, pp. 311-533. | MR 2099037 | Zbl pre05177023

[38] Zumbrun K., Planar stability criteria for viscous shock waves of systems with real viscosity, in: Hyperbolic Systems of Balance Laws, Springer Lecture Notes, 2006, in press.

[39] Zumbrun K., Serre D., Viscous and inviscid stability of multidimensional planar shock fronts, Indiana Univ. Math. J. 48 (1999) 937-992. | MR 1736972 | Zbl 0944.76027