# Global in Time Stability of Steady Shocks in Nozzles

Jeffrey Rauch^{[1]}; Chunjing Xie^{[1]}; Zhouping Xin^{[2]}

- [1] Department of Mathematics University of Michigan 530 Church Street Ann Arbor, MI 48109 USA
- [2] The Institute of Mathematical Sciences and department of mathematics The Chinese University of Hong Kong Hong Kong

Séminaire Laurent Schwartz — EDP et applications (2011-2012)

- Volume: 2011-2012, page 1-11
- ISSN: 2266-0607

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topRauch, Jeffrey, Xie, Chunjing, and Xin, Zhouping. "Global in Time Stability of Steady Shocks in Nozzles." Séminaire Laurent Schwartz — EDP et applications 2011-2012 (2011-2012): 1-11. <http://eudml.org/doc/251175>.

@article{Rauch2011-2012,

abstract = {We prove global dynamical stability of steady transonic shock solutions in divergent quasi-one-dimensional nozzles. One of the key improvements compared with previous results is that we assume neither the smallness of the slope of the nozzle nor the weakness of the shock strength. A key ingredient of the proof are the derivation a exponentially decaying energy estimates for a linearized problem.},

affiliation = {Department of Mathematics University of Michigan 530 Church Street Ann Arbor, MI 48109 USA; Department of Mathematics University of Michigan 530 Church Street Ann Arbor, MI 48109 USA; The Institute of Mathematical Sciences and department of mathematics The Chinese University of Hong Kong Hong Kong},

author = {Rauch, Jeffrey, Xie, Chunjing, Xin, Zhouping},

journal = {Séminaire Laurent Schwartz — EDP et applications},

keywords = {transonic shock solutions; inviscid compressible isentropic Euler equations},

language = {eng},

pages = {1-11},

publisher = {Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique},

title = {Global in Time Stability of Steady Shocks in Nozzles},

url = {http://eudml.org/doc/251175},

volume = {2011-2012},

year = {2011-2012},

}

TY - JOUR

AU - Rauch, Jeffrey

AU - Xie, Chunjing

AU - Xin, Zhouping

TI - Global in Time Stability of Steady Shocks in Nozzles

JO - Séminaire Laurent Schwartz — EDP et applications

PY - 2011-2012

PB - Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique

VL - 2011-2012

SP - 1

EP - 11

AB - We prove global dynamical stability of steady transonic shock solutions in divergent quasi-one-dimensional nozzles. One of the key improvements compared with previous results is that we assume neither the smallness of the slope of the nozzle nor the weakness of the shock strength. A key ingredient of the proof are the derivation a exponentially decaying energy estimates for a linearized problem.

LA - eng

KW - transonic shock solutions; inviscid compressible isentropic Euler equations

UR - http://eudml.org/doc/251175

ER -

## References

top- Courant, R. and Friedrichs, K.O., Supersonic Flow and Shock Waves, Springer-Verlag, 1948. Zbl0365.76001MR29615
- Elling, V., Regular reflection in self-similar potential flow and the sonic criterion. Commun. Math. Anal. 8 (2010), no. 2, 22-69. Zbl1328.76038MR2569954
- Elling, V., Instability of strong regular reflection and counterexamples to the detachment criterion. SIAM J. Appl. Math. 70 (2009), no. 4, 1330-1340. Zbl05779410MR2563517
- Elling, V., Counterexamples to the sonic criterion. Arch. Ration. Mech. Anal. 194 (2009), no. 3, 987-1010. Zbl1255.76049MR2563630
- P. Embid, J. Goodman, and A. Majda,Multiple steady states for $1$-D transonic flow, SIAM J. Sci. Statist. Comput. 5 (1984), no. 1, 21–41. Zbl0573.76055MR731879
- L. C. Evans, Partial differential equations, Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 1998. Zbl0902.35002MR1625845
- S.-Y. Ha, ${L}^{1}$ stability for systems of conservation laws with a nonresonant moving source, SIAM J. Math. Anal. 33 (2001), no. 2, 411–439. Zbl1002.35086MR1857977
- S.-Y. Ha and T. Yang, ${L}^{1}$ stability for systems of hyperbolic conservation laws with a resonant moving source, SIAM J. Math. Anal. 34 (2003), no. 5, 1226–1251 Zbl1036.35128MR2001667
- T. Kato, Perturbation theory for linear operators, Reprint of the 1980 edition, Classics in Mathematics, Springer-Verlag, Berlin, 1995. Zbl0836.47009MR1335452
- P. D. Lax, Functional analysis, Pure and Applied Mathematics (New York), Wiley-Interscience [John Wiley & Sons], New York, 2002. Zbl1009.47001MR1892228
- Ta Tsien Li and Wen Ci Yu, Boundary value problems for quasilinear hyperbolic systems, Duke University Mathematics Series, V. Duke University, Mathematics Department, Durham, NC, 1985. Zbl0627.35001MR823237
- Wen-Ching Lien, Hyperbolic conservation laws with a moving source, Comm. Pure Appl. Math. 52 (1999), no. 9, 1075–1098. Zbl0932.35142MR1692156
- Tai-Ping Liu, Transonic gas flow in a duct of varying area, Arch. Rational Mech. Anal. 80 (1982), no. 1, 1–18. Zbl0503.76076MR656799
- Tai-Ping Liu, Nonlinear stability and instability of transonic flows through a nozzle, Comm. Math. Phys. 83 (1982), no. 2, 243–260. Zbl0576.76053MR649161
- Tai-Ping Liu, Nonlinear resonance for quasilinear hyperbolic equation, J. Math. Phys. 28 (1987), no. 11, 2593–2602. Zbl0662.35068MR913412
- Tao Luo, Jeffrey Rauch, Chunjing Xie, and Zhouping Xin, Stability of transonic shock solutions for Euler-Poisson equations, Archive Rational Anal. Mech., to appear, arXiv:1008.0378.
- Andrew Majda, The existence of multidimensional shock fronts, Mem. Amer. Math. Soc. 43 (1983), no. 281. Zbl0517.76068MR699241
- Guy Métivier, Stability of multidimensional shocks, Advances in the theory of shock waves, 25–103, Progr. Nonlinear Differential Equations Appl., 47, Birkhuser Boston, Boston, MA, 2001. Zbl1017.35075MR1842775
- Jeffrey Rauch, Qualitative behavior of dissipative wave equations on bounded domains, Arch. Rational Mech. Anal. 62 (1976), no. 1, 77–85. Zbl0335.35062MR404864
- Jeffrey Rauch and Frank Massey, Differentiability of solutions to hyperbolic initial-boundary value problems, Trans. Amer. Math. Soc. 189 (1974), 303–318. Zbl0282.35014MR340832
- Jeffrey Rauch and Michael Taylor, Exponential decay of solutions to hyperbolic equations in bounded domains, Indiana Univ. Math. J. 24 (1974), 79–86. Zbl0281.35012MR361461
- Zhouping Xin and Huicheng Yin, The transonic shock in a nozzle, 2-D and 3-D complete Euler systems, J. Differential Equations 245 (2008), no. 4, 1014–1085. Zbl1165.35031MR2427405

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