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Mat. Sb., 2009, Volume 200, Number 2, Pages 3–30 (Mi msb6363)  

This article is cited in 12 scientific papers (total in 12 papers)

Stability of a supersonic flow about a wedge with weak shock wave

A. M. Blokhin, D. L. Tkachev

Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences

Abstract: It is known that the problem of finding the streamlines of a stationary supersonic flow of a nonviscous nonheat-conducting gas in thermodynamical equilibrium past an infinite plane wedge (with a sufficiently small angle at the vertex) in theory has two solutions: a strong shock wave solution (the velocity behind the front of the shock wave is subsonic) and a weak shock wave solution (the velocity behind the front of the shock wave is generally speaking supersonic). In the present paper it is shown for a linear approximation to this problem that the weak shock wave solution is asymptotically stable in the sense of Lyapunov. Moreover, it is shown that for initial data with compact support the solution of the mixed linear problem converges in finite time to the zero solution. In the case of linear approximation these results complete the verification of the well-known Courant-Friedrichs conjecture that the strong shock wave solution is unstable, whereas the weak shock wave solution is asymptotically stable in the sense of Lyapunov.
Bibliography: 39 titles.

Keywords: weak shock wave, asymptotic stability (in the sense of Lyapunov).
Author to whom correspondence should be addressed

DOI: https://doi.org/10.4213/sm6363

Full text: PDF file (679 kB)
References: PDF file   HTML file

English version:
Sbornik: Mathematics, 2009, 200:2, 157–184

Bibliographic databases:

UDC: 517.956.3
MSC: 76J20, 34D20
Received: 20.05.2008 and 26.11.2008

Citation: A. M. Blokhin, D. L. Tkachev, “Stability of a supersonic flow about a wedge with weak shock wave”, Mat. Sb., 200:2 (2009), 3–30; Sb. Math., 200:2 (2009), 157–184

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Askari S., Shojaeefard M.H., Goudarzi K., “Numerical and analytical solution of compressible flow over double wedge and biconvex airfoils”, Engineering Computations, 28:3-4 (2011), 441–471  crossref  zmath  isi  elib  scopus
    2. D. Tkachev, A. Blokhin, “Courant-Friedrichs' hypothesis and stability of the weak shock wave satisfying the Lopatinski condition”, Open Journal of Applied Sciences, 3:1 (2013), 79–83  crossref  mathscinet
    3. M. Darwish, F. Moukalled, “A fully coupled Navier–Stokes solver for fluid flow at all speeds”, Numerical Heat Transfer, Part B: Fundamentals, 65:5 (2014), 410–444  crossref  adsnasa  isi  scopus
    4. A. N. Kraiko, K. S. P'yankov, Ye. A. Yakovlev, “The flow of a supersonic ideal gas with “weak” and “strong” shocks over a wedge”, J. Appl. Math. Mech., 78:4 (2014), 318–330  crossref  mathscinet  zmath  isi  elib  elib  scopus
    5. A. M. Blokhin, D. L. Tkachev, “Stability of a supersonic flow over a wedge containing a weak shock wave satisfying the Lopatinski condition”, J. Hyberbolic Differ. Equ., 11:2 (2014), 215–248  crossref  mathscinet  zmath  isi  scopus
    6. Blokhin A.M., Tkachev D.L., “Stability condition for strong shocks in flows over an infinite planar wedge satisfying the Lopatinski condition”, J. Hyberbolic Differ. Equ., 12:4 (2015), 817–847  crossref  mathscinet  zmath  isi  scopus
    7. A. M. Blokhin, D. L. Tkachev, “Stability of a supersonic flow past a wedge with adjoint weak neutrally stable shock wave”, Siberian Adv. Math., 27:2 (2017), 77–102  mathnet  crossref  crossref  elib
    8. Blokhin A., Tkachev D., “Linear stability of a weak shock wave appearing in flow over an infinite plane wedge (Lopatinski condition is fulfilled on the shock)”, INTERNATIONAL CONFERENCE ON THE METHODS OF AEROPHYSICAL RESEARCH (ICMAR 2016): Proceedings of the 18th International Conference on the Methods of Aerophysical Research (Perm, Russia, 27 June–3 July 2016), AIP Conference Proceedings, 1770, ed. Fomin V., Amer Inst Physics, 2016, 030080  crossref  isi  scopus  scopus
    9. Tkachev D.L., Blokhin A.M., “The problem of flow about infinite plane wedge with inviscous non-heat-conducting gas. Linear stability of a weak shock wave”, 2016 Days on Diffraction (DD) (St.Petersburg, Russia), eds. Motygin O., Kiselev A., Kapitanova P., Goray L., Kazakov A., Kirpichnikova A., IEEE, 2016, 410–415  crossref  isi  scopus  scopus
    10. Blokhin A.M., Tkachev D.L., “Local Well-Posedness in the Problem of Flow About Infinite Plane Wedge With Inviscous Non-Heat-Conducting Gas”, Proceedings of the International Conference Days on Diffraction (Dd) 2017, eds. Motygin O., Kiselev A., Goray L., Suslina T., Kazakov A., Kirpichnikova A., IEEE, 2017, 62–67  isi
    11. E. V. Semenko, T. I. Semenko, “The shock front asymptotics in the linear problem of shock wave”, Sib. elektron. matem. izv., 15 (2018), 950–970  mathnet  crossref
    12. A. M. Blokhin, D. L. Tkachev, A. V. Yegitov, “Local solvability of the problem of the van der Waals gas flow around an infinite plane wedge in the case of a weak shock wave”, Siberian Math. J., 59:6 (2018), 960–982  mathnet  crossref  crossref  isi  elib
  • Математический сборник Sbornik: Mathematics (from 1967)
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