Lecture notes : Stability of Noncharacteristic Viscous Boundary Layers

Guy Métivier[1]

  • [1] IMB Université de Bordeaux I, 33405 Talence Cedex

Journées Équations aux dérivées partielles (2006)

  • page 1-82
  • ISSN: 0752-0360

How to cite

top

Métivier, Guy. "Lecture notes : Stability of Noncharacteristic Viscous Boundary Layers." Journées Équations aux dérivées partielles (2006): 1-82. <http://eudml.org/doc/10619>.

@article{Métivier2006,
affiliation = {IMB Université de Bordeaux I, 33405 Talence Cedex},
author = {Métivier, Guy},
journal = {Journées Équations aux dérivées partielles},
language = {eng},
month = {6},
pages = {1-82},
publisher = {Groupement de recherche 2434 du CNRS},
title = {Lecture notes : Stability of Noncharacteristic Viscous Boundary Layers},
url = {http://eudml.org/doc/10619},
year = {2006},
}

TY - JOUR
AU - Métivier, Guy
TI - Lecture notes : Stability of Noncharacteristic Viscous Boundary Layers
JO - Journées Équations aux dérivées partielles
DA - 2006/6//
PB - Groupement de recherche 2434 du CNRS
SP - 1
EP - 82
LA - eng
UR - http://eudml.org/doc/10619
ER -

References

top
  1. Bardos C., Brezis D. and Brezis H., Perturbations singulières et prolongement maximaux d’opérateurs positifs, Arch.Rational Mech. Anal., 53 (1973), pp 69–100. Zbl0281.47028
  2. Bardos C. and Rauch J., Maximal positive boundary value problems as limits of singular perturbation problems, Trans. Amer.Math.Soc., 270 (1982), pp 377–408. Zbl0485.35010MR645322
  3. Bony J-M., Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires, Ann. Sc. E.N.S. Paris, 14 (1981) pp 209–246. Zbl0495.35024MR631751
  4. Chazarain J. and Piriou, A., Introduction to the Theory of Linear Partial Differential Equations, North Holland, Amsterdam, 1982. Zbl0487.35002MR678605
  5. Gardner R. and Zumbrun K., The gap lemma and geometric criteria instability of viscous shock profiles, Comm. Pure Appl. Math, 51 (1998), pp 797–855. Zbl0933.35136MR1617251
  6. Gilbarg, D., The existence and limit behavior of the one-dimensional shock layer, Amer. J. Math. 73. 1951, 256-274. Zbl0044.21504MR44315
  7. Gisclon M. Étude des conditions aux limites pour un système strictement hyperbolique, via l’approximation parabolique. J. Math. Pures Appl., 75 (1996), pp 485–508. Zbl0869.35061
  8. Gisclon M. and Serre D., Étude des conditions aux limites pour un système strictement hyberbolique via l’approximation parabolique. C. R. Acad. Sci. Paris SÈr. I Math., 319 (1994), pp 377–382. Zbl0808.35075
  9. Goodman, J., Nonlinear asymptotic stability of viscous shock profiles for conservation laws, Arch. Rational Mech. Analysis 95. (1986), pp 325–344. Zbl0631.35058MR853782
  10. Goodman J. and Xin Z. Viscous limits for piecewise smooth solutions to systems of conservation laws, Arch. Rational Mech. Analysis 121 (1992), pp 235–265. Zbl0792.35115MR1188982
  11. Grenier E. and Guè 0., Boundary layers for viscous perturbations of noncharacteristic quasilinear hyperbolic problems, J.Diff.Equ., 143 (1998) pp 110–146. Zbl0896.35078MR1604888
  12. Grenier E. and Rousset F., Stability of one dimensional boundary layers by using Green’s functions, Comm. Pure Appl. Math. 54 (2001), pp 1343–1385. Zbl1026.35015
  13. Guès O., Perturbations visqueuses de problèmes mixtes hyperboliques et couches limites, Ann.Inst.Fourier, 45 (1995) pp 973–1006. Zbl0831.34023MR1359836
  14. Guès O., Problème mixte hyperbolique quasilinéaire caractéristique, Comm. in Part.Diff.Equ., 15 (1990), pp 595–645. Zbl0712.35061MR1070840
  15. Guès O., Métivier G., Williams M., and Zumbrun, K., Multidimensional viscous shocks I: degenerate symmetrizers and long time stability, Journal of the AMS, 18 (2005), pp 61-120 Zbl1058.35163MR2114817
  16. Guès O., Métivier G., Williams M., and Zumbrun, K., Multidimensional viscous shocks II: the small viscosity problem, Comm. Pure and Appl. Math., 57 (2004) pp 141–218. Zbl1073.35162MR2012648
  17. Guès O., Métivier G., Williams M., and Zumbrun, K., Existence and stability of multidimensional shock fronts in the vanishing viscosity limit, Arch.Rat.Mech.Anal., 175 (2005) pp 151–244. Zbl1072.35122MR2118476
  18. Guès O., Métivier G., Williams M., and Zumbrun, K., Navier-Stokes regularization of multidimensional Euler shocks, Ann. Scient. Ec. Norm. Sup., to appear. Zbl1173.35082MR2224659
  19. Guès O., Métivier G., Williams M., and Zumbrun, K., Nonclassical multidimensional viscous and inviscid shocks, preprint Zbl1162.35050
  20. Guès O., Métivier G., Williams M., and Zumbrun, K., Viscous Boundary Value Problems for Symmetric Systems with Variable Multiplicities, preprint Zbl1138.35052
  21. Guès O., Métivier G., Williams M., and Zumbrun, K., Uniform stability estimates for constant-coefficient symmetric hyperbolic boundary value problems, Comm. in Partial Diff. Equ., to appear Zbl1133.35404MR2334823
  22. Guès O., Métivier G., Williams M., and Zumbrun, K., Stability of noncharacteristic boundary layers for the compressible Navier-Stokes and MHD equations, in preparation. Zbl1217.35136
  23. Gues O. and Williams M., Curved shocks as viscous limits: a boundary problem approach, Indiana Univ. Math. J., 51 (2002), pp 421-450. Zbl1046.35072MR1909296
  24. Hoff D. and Zumbrun K., Multi-dimensional diffusion waves for the Navier-Stokes equations of compressible flow. Indiana Univ. Math. J., 44 (1995), pp 603–676. Zbl0842.35076MR1355414
  25. Kawashima S. and Shizutz Y., Systems of equations of hyperbolic-parabolic type, with applications to the discrete Boltzmann equations, Hokkaido Math.J., 14 (1985) pp 249–275. Zbl0587.35046MR798756
  26. Kawashima S. and Shizutz Y, On the normal form of the symmetric hyperbolic-parabolic systems associated with the conservation laws, Tohoku Math.J., 40 (1988) pp 449–464 Zbl0699.35171MR957056
  27. Kreiss H.O., Initial boundary value problems for hyperbolic systems, Comm. Pure Appl. Math., 23 (1970), pp. 277-298. Zbl0193.06902MR437941
  28. Lions J.L., Perturbations singulières dans les problèmes aux limites et en contrôle optimal, Lectures Notes in Math., 323, Sringer Verlag, 1973. Zbl0268.49001MR600331
  29. Majda A., The stability of Multidimensional Shock Fronts, Mem. Amer. Math. Soc., n o 275, 1983. Zbl0506.76075MR683422
  30. Majda A., The Existence of Multidimensional Shock Fronts, Mem. Amer. Math. Soc., n o 281, 1983. Zbl0517.76068MR699241
  31. Majda A., and Osher S., Initial-boundary value problems for hyperbolic equations with uniformly characteristic boundary, Comm. Pure Appl. Math., 28 (1975), pp 607-676. Zbl0314.35061MR410107
  32. Majda, A. and Pego, R., Stable viscosity matrices for systems of conservation laws, J. Diff. Eq., 56. (1985), pp 229-262. Zbl0512.76067MR774165
  33. Malgrange B., Ideals of differentiable functions, Oxford Univ. Press, 1966. Zbl0177.17902MR212575
  34. Métivier G., Stability of multidimensional weak shocks, Comm., Partial Diff. Eq., 15 (1990), pp 983-1028. Zbl0711.35078MR1070236
  35. Métivier G., The Block Structure Condition for Symmetric Hyperbolic Problems, Bull. London Math.Soc., 32 (2000), pp 689–702 Zbl1073.35525MR1781581
  36. Métivier G, Stability of multidimensional shocks. Advances in the theory of shock waves, 25–103, Progr. Nonlinear Differential Equations Appl., 47, Birkhäuser Boston, Boston, MA, 2001. Zbl1017.35075MR1842775
  37. Métivier G., Small Viscosity and Boundary Layer Methods, Birkhäuser, Boston 2004. Zbl1133.35001MR2151414
  38. Métivier G. and - Zumbrun K., Viscous Boundary Layers for Noncharacteristic Nonlinear Hyperbolic Problems, Memoirs AMS, 826 (2005). Zbl1074.35066
  39. Métivier G. and - Zumbrun K., Hyperbolic Boundary Value Problems for Symmetric Systems with Variable Multiplicities, J. Diff. Equ., 211 (2005) pp 61-134. Zbl1073.35155MR2121110
  40. Métivier G. and - Zumbrun K., Symmetrizers and continuity of stable subspaces for parabolic–hyperbolic boundary value problems. J. Dis. and Cont. Dyn. Sys., 11 (2004) pp 205-220. Zbl1102.35332MR2073953
  41. Pego R., Stable viscosities and shock profiles for systems of conservation laws, Trans. A.M.S., 282 (1984), pp 749-763. Zbl0512.76068MR732117
  42. Plaza R. and Zumbrun K., An Evans function approach to spectral stability of small-amplitude shock profiles, J. Disc. and Cont. Dyn. Sys. 10. (2004), pp 885–924. Zbl1058.35164MR2073940
  43. Ralston J., Note on a paper of Kreiss, Comm. Pure Appl. Math., 24 (1971), pp 759–762. Zbl0215.16802MR606239
  44. Rauch J., L 2 is a continuable initial condition for Kreiss’ mixed problems, Comm.Pure and Appl.Math., 25 (1972) pp 265–285. Zbl0226.35056
  45. Rauch J., Symmetric positive systems with boundary characteristic of constant multiplicity, Trans. Amer.Math.Soc, 291 (1985), pp 167–185. Zbl0549.35099MR797053
  46. Rousset F., Inviscid boundary conditions and stability of viscous boundary layers, Asympt.Anal., 26 (2001) pp 285–306. Zbl0977.35081MR1844545
  47. Rousset F., Viscous approximation of strong shocks of systems of conservation laws. SIAM J. Math. Anal. 35 (2003), pp 492–519. Zbl1052.35128MR2001110
  48. Shiota M., Nash Manifolds, Lectures Notes in Mathematics, 1269, Springer Verlag. Zbl0629.58002MR904479
  49. Serre D., Sur la stabilité des couches limites de viscosité. Ann. Inst. Fourier (Grenoble) 51 (2001), pp 109–130. Zbl0963.35009MR1821071
  50. Zumbrun K., Multidimensional stability of planar viscous shock waves. Advances in the theory of shock waves, 307–516, Progr. Nonlinear Differential Equations Appl., 47, Birkhäuser Boston, Boston, MA, 2001. Zbl0989.35089MR1842778
  51. Zumbrun K. Stability of large-amplitude shock waves of compressible Navier–Stokes equations. For Handbook of Fluid Mechanics III, S.Friedlander, D.Serre ed., Elsevier North Holland (2004). Zbl1222.35156MR2099037
  52. Zumbrun K. and Howard P., Pointwise semigroup methods and stability of viscous shock waves. Indiana Univ. Math. J., 47 (1998), pp 741–871. Zbl0928.35018MR1665788
  53. Zumbrun K. and Serre D., Viscous and inviscid stability of multidimensional planar shock fronts , Indiana Univ. Math. J., 48 (1999), pp 937–992. Zbl0944.76027MR1736972

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.