×

zbMATH — the first resource for mathematics

Stability of one-dimensional boundary layers by using Green’s functions. (English) Zbl 1026.35015
The authors investigate the stability of one-dimensional boundary layers of parabolic systems as the viscosity goes to 0 in the noncharacteristic case and prove that spectral stability implies linear and nonlinear stability of approximate solutions. In particular, they replace the smallness condition obtained by the energy method of M. Gisclon–D. Serre and E. Grenier–O. Guès by a weaker spectral condition.
Reviewer: Jiaqi Mo (Wuhu)

MSC:
35B35 Stability in context of PDEs
35B25 Singular perturbations in context of PDEs
35L65 Hyperbolic conservation laws
35K55 Nonlinear parabolic equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Stability of semi-discrete shock profiles by means of an Evans function in infinite dimension. Preprint, 2000.
[2] Benzoni-Gavage, SIAM J Math Anal 32 pp 929– (2001)
[3] Bultelle, SIAM J Numer Anal 35 pp 2272– (1998)
[4] Case, Phys Fluids 3 pp 143– (1960)
[5] Dichotomies in stability theory. Lecture Notes in Mathematics, 629. Springer, Berlin-New York, 1978. · Zbl 0376.34001
[6] ; Linear instability implies nonlinear instability for various boundary layers. Preprint, 2000.
[7] ; Hydrodynamic stability. Cambridge Monographs on Mechanics and Applied Mathematics. Cambridge University Press, Cambridge-New York, 1981.
[8] ; ; Two-dimensional perturbations of a flow with constant shear of a stratified fluid. Publication No. 1. Institute for Weather and Climate Research, the Norwegian Academy of Sciences and Letters. 1953.
[9] Gardner, Comm Pure Appl Math 51 pp 797– (1998)
[10] Gisclon, C R Acad Sci Paris Sér I Math 319 pp 377– (1994)
[11] Godillon, Phys D 148 pp 289– (2001)
[12] Grenier, Comm Pure Appl Math 53 pp 1067– (2000)
[13] Grenier, J Differential Equations 143 pp 110– (1998)
[14] Guès, Ann Inst Fourier (Grenoble) 45 pp 973– (1995) · Zbl 0831.34023
[15] Perturbation theory for linear operators. Springer, Berlin, 1995. Reprint of the 1980 edition.
[16] Kreiss, Comm Pure Appl Math 51 pp 1397– (1998)
[17] ; Cours de physique. Éditions Mir, Moscow. · JFM 05.0578.02
[18] Lilly, J Atmos Sci pp 481– (1966)
[19] Liu, Comm Pure Appl Math 50 pp 1113– (1997)
[20] Rousset, Asymptot Anal
[21] Serre, Ann Inst Fourier (Grenoble) 51 pp 109– (2001) · Zbl 0963.35009
[22] Discrete shock profiles and their stability. Hyperbolic problems: theory, numerics, applications. Vol. II. Proceedings of the 7th International Conference held in Zürich, February 1998, 843-854.
[23] International Series of Numerical Mathematics, 130. Birkhäuser, Basel, 1999.
[24] Yu, Arch Ration Mech Anal 146 pp 275– (1999)
[25] Multidimensional stability of planar viscous shock waves. Notes of TMR Summer School, Köchel am See, Germany, 1999.
[26] Zumbrun, Indiana Univ Math J 47 pp 741– (1998)
[27] Zumbrun, Indiana Univ Math J 48 pp 937– (1999)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.